Livro texto de Cálculo IIA UFF - Cristiane

Prévia do material em texto

n � 2
n
n � 3
n � 3
n � 3
x
f [a, b]
f
3, 7, 15 30
2
f
[a, b] x
n
[t0, t1], [t1, t2], · · · , [tn�1, b] {f(t⇤1), f(t⇤2), · · · , t⇤n}
{a = t0 < t⇤1 < t1 < · · · < tn�1 < t⇤n < tn = b}.
n = 6
P = ({t0, t1, · · · , tn�1, tn}, {t⇤1, · · · , t⇤n�1, t⇤n})
||P||
[ti, ti+1]
f : [a, b] ! R
x Z b
a
f(x)dx = lim
||P||!0
nX
i=1
f(t⇤i )(ti � ti�1),
f
[a, b]
•
f
x
x
•
f : [a, b] ! R
Z 1
0
x2 dx
f(x) = x2
[0, 1]
{Pn}n�1 ||Pn||! 0 n!1
Pn = ({0, 1
n
,
2
n
, · · · , n� 1
n
, 1}, { 1
n
,
2
n
, · · · , n� 1
n
, 1}).
||Pn|| = 1n
Pn
Pn
i=1(
i
n)
2( 1n).
n
nX
i=1
i2 =
n(n+ 1)(2n+ 1)
6
,
Z 1
0
f(x)dx = lim
n!0
1
n3
nX
i=1
i2 =
1
3
.
⇤
f : [a, b] ! R g : [a, b] ! R
Z b
a
f(x) + g(x) dx =
Z b
a
f(x) dx+
Z b
a
g(x) dx .
Z b
a
kf(x) dx = k
Z b
a
f(x) dx k 2 R
f(x)  g(x) x 2 [a, b]
Z b
a
f(x) dx 
Z b
a
g(x) dx .
Z b
a
f(x) dx =
Z c
a
f(x) dx+
Z b
c
f(x) dx c
a < c < b
m,M 2 R m  f(x) M x 2 [a, b]
m(b� a) 
Z b
a
f(x) dx M(b� a) .
Z a
a
f(x) dx = 0
Z 1
�1
p
1� x2dx
f [0, 1]
f(x) =
8<:0 x ,1 x ,
{t⇤i }ni=1
Z 1
0
x3 dx =
1
4
nX
k=1
k3 =
hn(n+ 1)
2
i2
.
Z 3
0
[5f(x) + 4g(x)] dx
Z 3
�2
f(x) dx = 7Z 0
�2
f(x) dx = 15
Z 3
0
g(x) dx = 10
2
p
3 
Z 1
�1
p
3 + x2 dx  4 ;
⇡
24 
Z ⇡
4
⇡
6
sen(x) dx 
p
2⇡
24
.
f : [a, b]! R [a, b] g : [a, b]! R
x [a, b] g(x) =
Z x
a
f(t) dt
f � 0 g(x)
g(x) f(t) t
t 2 [a, x] g(a) =
Z a
a
f(t) dt = 0
f(t) t 2 [a, x]
g(0) g(1) g(3/2) g(2) g(3) g(x) =
Z x
0
f(t) dt
f(t) =
8><>:
t2, se 0  t  1;
1, se 1 < t < 2
3� t, se 2  t  3
g(0) =
Z 0
0
f(t) dt = 0 g(1) =
Z 1
0
f(t) dt =
Z 1
0
t2 dt = 1/3
g(3/2) =
Z 3/2
0
f(t) dt =
Z 1
0
f(t) dt+
Z 3/2
1
f(t) dt
=
Z 1
0
t2 dt+
Z 3/2
1
1 dt = 1/3 + 1/2 = 5/6 ,
g(2) =
Z 2
0
f(t) dt =
Z 1
0
f(t) dt+
Z 2
1
f(t) dt =
Z 1
0
t2 dt+
Z 2
1
1 dt = 1/3+1 = 4/3 ,
g(3) =
Z 2
0
f(t) dt+
Z 3
2
f(t) dt = 4/3 +
Z 3
2
3� t dt = 4/3 + 1/2 = 11/6 ,
⇤
f(t)
d
dx
Z x
0
t dt = x 8x � 0
y = t
t t 2 [0, x]
Z x
0
t dt =
x.x
2
=
x2
2
x
d
dx
Z x
0
t dt =
d
dx
✓
x2
2
◆
= x. ⇤
f
I F I
f F 0(x) = f(x) 8x 2 I
F G
f I F (x) = G(x) + C
8x 2 I CZ x
0
t dt f(x) = x
f � 0
g(x)
g0(x) = lim
h!0
g(x+ h)� g(x)
h
.
h > 0
g(x+ h)� g(x) =
Z x+h
a
f(t) dt�
Z x
a
f(t) dt
=
Z x
a
f(t) dt+
Z x+h
x
f(t) dt�
Z x
a
f(t) dt
=
Z x+h
x
f(t) dt .
f t t 2 [x, x + h]
h! 0
h ⇡ 0
f(x)
g0(x) = lim
h!0
g(x+ h)� g(x)
h
⇡ lim
h!0
f(x).h
h
= lim
h!0
f(x) = f(x) .
h < 0
f
g(x) =
Z x
a
f(t) dt [a, b] (a, b)
g0(x) = f(x) 8x 2 (a, b)
Z b
a
f(x) dx = F (b)� F (a) F f
•
a b g0(x) = f(x)
8x 2 [a, b] g0(a) g x = a g0(b)
•
F
F (b) � F (a)
F (b)� F (a) = F (x)
���b
a
.
•
• d
dx
Z x
a
f(t) dt = f(x)
f fZ x
a
d
dt
f(t) dt = f(x)� f(a)
f f(a)
•
Z 4
1
1
x
dx
Z 1
0
1
x2 + 1
dx
Z 2
�1
3x4 + 1 dxZ ⇡
0
sen x dx
Z 4
1
�1p
t
dt
Z 1
0
es ds
R
y = x4 Ox x 2 [0, 2]
y = cos x Ox x 2 [0, ⇡/2]
y = 1 y = cos x x 2 [0, ⇡/2]
xZ x
1
2
t
dt
Z x
0
es
2
ds
Z 1
x
u2
u4 + 2u2 + 3
du
Z
xa dx =
xa+1
a+ 1
+ C 8a 2 R, a 6= �1 ,
Z
ex dx = ex + C ,Z
cosx dx = senx+ C ,
Z
sen x dx = � cosx+ C ,Z
sec2 x dx = tg x+ C ,
Z
1
x
dx = ln |x|+ C ,Z
1
1 + x2
dx = arctg x+ C ,
Z
1p
1� x2 dx = arcsen x+ C .
Z
cf(x) dx = c
Z
f(x) dx c
Z
[f(x) + g(x)] dx =
Z
f(x) dx+
Z
g(x) dx
Z
sec2 x+ 3x4 dx = tg x+
3
5
x5 + C .
R f(x) =
p
x x
x 2 [1, 2]
R
RZ 2
1
p
x dx =
Z 2
1
x
1
2 dx =
x3/2
3/2
���2
1
=
2
3
[23/2 � 13/2] = 2
3
[2
p
2� 1] .
⇤
R
• f(x)  0 8x 2 [a, b] f
x [a, b] A(R) = A(R⇤) =
Z b
a
�f(x) dx = �
Z b
a
f(x) dx
• f [a, b] [a, b]
f
f [a, b]
f �f
f
�
Z b
a
f(x) dx
A(R) =
Z x1
a
f(x) dx �
Z x2
x1
f(x) dx
+
Z x3
x2
f(x) dx�
Z b
x3
f(x) dx
y = senx x = 0 x = 2⇡
Z 2⇡
0
sen x dx
y = senx x [0, 2⇡]
Z 2⇡
0
sen x dx = � cosx
���2⇡
0
= � cos 2⇡ + cos 0 = �1 + 1 = 0
R
sen x [⇡, 2⇡]
y = � sen x [⇡, 2⇡]
x
Z ⇡
0
sen x dx+
Z 2⇡
⇡
� sen x dx = � cosx
���⇡
0
� (� cosx)
���2⇡
⇡
=
� cos ⇡ + cos 0 + cos 2⇡ � cos ⇡ = �(�1) + 1 + 1� (�1) = 4
Z 2⇡
⇡
sen x dx
x
Z 2⇡
0
sen x dx =
Z ⇡
0
sen x dx
+
Z 2⇡
⇡
sen x dx
xZ ⇡
0
sen x dx�
Z 2⇡
⇡
sen x dx 6=
Z 2⇡
0
sen x dx.
⇤
f(x) = x3�x2�6x
x
f(x) = x3 � x2 � 6x x
f(x) = x3 � x2 � 6x = 0
f(x) = x3�x2�6x = 0, x(x�3)(x+2) = 0, x = 0 ou x = 3 ou x = �2 .
f
A(R) =
Z 0
�2
x3 � x2 � 6x dx�
Z 3
0
x3 � x2 � 6x dx
=

x4
4
� x
3
3
� 6x
2
2
� ���0
�2
�

x4
4
� x
3
3
� 6x
2
2
� ���3
0
= �4� 8/3 + 12� 81/4 + 27/3 + 27 = 167/12.
⇤
y = x2 y = x
x2 = x , x = 0 ou x = 1
y = x x y = x2
x x 2 [0, 1]
A(R) =
Z 1
0
x dx�
Z 1
0
x2 dx =
x2
2
���1
0
� x
3
3
���1
0
= 1/2� 1/3 = 1/6.
R
⇤
f g [a, b]
f(x) = g(x) [a, b]
f(x) � g(x)
a b
Z b
a
|f(x) � g(x)| dx
y = senx y = sen 2x
x 2 [0, ⇡]
[0, ⇡] sen x = sen 2x, sen x = 2 senx cosx,
sen x = 0 ou cosx = 1/2, x = 0 ou x = ⇡ ou x = ⇡/3
A(R) =
Z ⇡/3
0
sen 2x� sen x dx+
Z ⇡
⇡/3
sen x� sen 2x dx
= [�cos 2x
2
+ cosx]
���⇡/3
0
+ [� cosx+ cos 2x
2
]
���⇡
⇡/3
= 1/4 + 1/2 + 1/2� 1 + 1 + 1/2 + 1/2 + 1/4 = 5/2.
⇤
a)
Z x
1
sen t2 dt , b)
Z 2
x
et
2
dt .
d
dx
Z x
1
sen t2 dt = senx2
d
dx
Z 2
x
et
2
dt =
d
dx

�
Z x
2
et
2
dt
�
= �ex2 ⇤
Z x
1
f(t) dt = x2 + 3x� 4 f(x)
f(x) =
d
dx
Z x
1
f(t) dt
�
=
d
dx
[x2 + 3x� 4] = 2x+ 3.
⇤
F (x) =
Z x
a
f(t) dt
Z u(x)
a
f(t) dt = F (u(x))
d
dx
Z u(x)
a
f(t) dt =
d
dx
F (u(x)) = F 0(u(x))u0(x) = f(u(x))u0(x),
f u
v
d
dx
Z a
v(x)
f(t) dt = � d
dx
Z v(x)
a
f(t) dt = �f(v(x))v0(x).
Z u(x)
v(x)
f(t) dt =
Z a
v(x)
f(t) dt+
Z u(x)
a
f(t) dt ,
a f
d
dx
Z u(x)
v(x)
f(t) dt = f(u(x))u0(x)� f(v(x))v0(x)
d
dx
Z x2
�x
et
3
dt
d
dx
Z x2
�x
et
3
dt = e(x
2)32x� e(�x)3(�1) = 2xex6 + e�x3
⇤
lim
x!0
Z x
�x
cos t5 dt
x
lim
x!0
Z x
�x
cos t5 dt
x
= lim
x!0
cosx5 � cos(�x)5(�1)
1
= lim
x!0
cosx5 + cosx5 = 2 .
⇤
Z 4
0
ex dx
Z 2
0
1
x2 + 1
dx
Z 2
�1
4x2 + 1 dxZ ⇡
⇡/2
sen x dx
Z 4
0
p
t dt
Z 5
1
1
s
ds
R
y = x3 + 2x2 � 3x x
y = x4 y = x x 2 [0, 2]
y = x� 2 x = y2
y = � sen x y = cos x x 2 [0, 2⇡]
x
f(x) =
Z x
1
cos t2 dt f(x) =
Z �x2
0
es
2
ds
x
Z 2x
�x3
cos 3
p
t dt
lim
x!0
Z x
�x
sen t2 dt
x3
lim
x!1
Z x2
x
et
2
dt
x� 1
Z
f(g(x))g0(x) dx,
g
f g
F f
Z
f(g(x))g0(x) dx =
Z
F 0(g(x))g0(x) dx
=
Z
d
dx
F (g(x)) dx = F (g(x)) + C.
x u u = g(x)
du = g0(x)dxZ
f(g(x))g0(x) dx =
Z
f(u)du = F (u) + C.
x u = g(x)
Z
f(g(x))g0(x) dx =
Z
f(u)du = F (u) + C = F (g(x)) + C,
x
dx du
x
x
Z
4
p
1 + 4x dx
u = 1 + 4x du = 4dxZ
4
p
1 + 4x dx =
Z p
u du =
u3/2
3/2+ C =
2
3
(1 + 4x)3/2 + C .
⇤
Z
u2eu
3
du
v = u3 dv = 3u2duZ
u2eu
3
du =
Z
1
3
ev dv =
1
3
ev + C =
eu
3
3
+ C .
⇤
Z
t(1 + t2)100 dt
u = 1 + t2 du = 2tdt) tdt = du
2Z
t(1+ t2)100 dt =
Z
u100
du
2
=
1
2
Z
u100 du =
1
2
u101
101
+C =
(1 + t2)101
202
+C .
⇤
Z
cos
p
xp
x
dx
u =
p
x du =
dx
2
p
x
) 2du = dxp
xZ
cos
p
xp
x
dx =
Z
cosu 2du = 2
Z
cosu du = 2 senu+ C = 2 sen
p
x+ C .
⇤
Z
tg x dx
tg x =
sen x
cosx
u = cos x du = � sen xdx) �du = senxdxZ
sen x
cosx
dx = �
Z
du
u
= � ln |u|+ C = � ln | cosx|+ C = ln | sec x|+ C .
⇤
Z
x
x2 + 6
dx
u = x2 + 6 du = 2xdxZ
x
x2 + 6
dx =
Z
1
2u
du =
1
2
ln |u|+ C = 1
2
ln(x2 + 6) + C .
⇤
Z
1
3x2 + 1
dx
u =
p
3x du =
p
3dxZ
1
(
p
3x)2 + 1
dx =
1p
3
Z
1
u2 + 1
du =
1p
3
arctg u+C =
1p
3
arctg(
p
3x)+C .
⇤
Z
1
x2 + 2x+ 5
dx
x2 + 2x+ 5 = (x+ 1)2 + 4 = 4
"✓
x+ 1
2
◆2
+ 1
#
,
u =
x+ 1
2
du =
1
2
dx
Z
1
x2 + 2x+ 5
dx =
1
4
Z
1✓
x+ 1
2
◆2
+ 1
dx =
1
2
Z
1
u2 + 1
du =
=
1
2
arctg u+ C =
1
2
arctg
✓
x+ 1
2
◆
+ C
⇤
Z
sec x dx
u = secx+ tg x du = [secxtgx+ sec2 x]dxZ
sec x dx =
Z
sec2 x+ sec x tg x
sec x+ tg x
dx =
Z
du
u
= ln | sec x+ tg x|+ C .
⇤
Z
4
p
1 + 4x dx =
2
3
(1 + 4x)3/2 + C ,
Z 2
1
4
p
1 + 4x dx =
2
3
(1+ 4x)3/2
���2
1
=
2
3
(1+ 8)3/2� 2
3
(1+ 4)3/2 = 18� 10
p
5
3
.
g0 f
g Z b
a
f(g(x))g0(x)dx =
Z g(b)
g(a)
f(u)du.
Z b
a
f(g(x))g0(x)dx = F (g(x))
���b
a
= F (g(b))� F (g(a)) ,
F fZ g(b)
g(a)
f(u)du = F (g(b))� F (g(a))
Z b
a
f(g(x))g0(x)dx =
Z g(b)
g(a)
f(u)du ,
⌅
Z 2
1
4
p
1 + 4x dx
u = 4x du = 4dx
x = 1) u = 1 + 4(1) = 5 x = 2) u = 1 + 4(2) = 9 .
Z 2
1
4
p
1 + 4x dx =
Z 9
5
p
u du =
2u3/2
3
���9
5
=
2
3
(93/2 � 53/2) = 18� 10
p
5
3
.
⇤
x
u
Z 0
�1
x
3x2 + 2
dx
u = 3x2 + 2 du = 6xdx
x = �1) u = 3(�1)2 + 2 = 5 x = 0) u = 3(0) + 2 = 2
Z 0
�1
x
3x2 + 2
dx =
Z 2
5
1
6
1
u
du =
1
6
lnu
���2
5
=
1
6
(ln 2� ln 5).
⇤
g(a)
g(b) g(b)  g(a)
y =
1
1� 2x x
x = �2 x = �1
A =
Z �1
�2
1
1� 2x dx
u = 1�2x du = �2dx) dx = �du
2
A =
Z �1
�2
1
1� 2x dx = �
1
2
Z 3
5
1
u
du = �1
2
lnu
���3
5
= �1
2
[ln 3�ln 5] = 1
2
ln
✓
5
3
◆
⇤
f [�a, a]
f f(�x) = f(x)
Z a
�a
f(x) dx = 2
Z a
0
f(x) dx
f f(�x) = �f(x)
Z a
�a
f(x) dx = 0
�a
a
Z a
�a
f(x) dx =
Z 0
�a
f(x) dx+
Z a
0
f(x) dx .
u = �x
du = �dx
Z 0
�a
f(x) dx = �
Z 0
a
f(�u) du .
f
Z 0
�a
f(x) dx = �
Z 0
a
f(�u) du = �
Z 0
a
f(u) du =
Z a
0
f(u) du,
Z a
�a
f(x) dx =
Z a
0
f(u) du+
Z a
0
f(x) dx = 2
Z a
0
f(x) dx ,
f
Z 0
�a
f(x) dx = �
Z 0
a
f(�u) du =
Z 0
a
f(u) du = �
Z a
0
f(u) du,
Z a
�a
f(x) dx =
Z 0
�a
f(x) dx+
Z a
0
f(x) dx = �
Z a
0
f(u) du+
Z a
0
f(x) dx = 0 .
⇤
Z ⇡
�⇡
sen3 x
cos8 x+ 3x2 + 1
dx
f(x) =
sen3 x
cos8 x+ 3x2 + 1
f(�x) = sen
3(�x)
cos8(�x) + 3(�x)2 + 1 =
� sen3 x
cos8 x+ 3x2 + 1
= �f(x) ,
x 2 [�⇡, ⇡]
Z ⇡
�⇡
sen3 x
cos8 x+ 3x2 + 1
dx = 0 .
⇤
Z
ln2 x
x
dx
Z
t
t4 + 1
dtZ
x
x4 + 2x2 + 3
dx
Z ⇡
0
sen ✓
2 cos2 ✓ + 1
d✓Z
sen 2x
3 cos2 x+ 1
dx
Z
1 + 2x
1 + 3x2
dxZ
x
p
x+ 3 dx
Z 1
�1
sp
3 + 2s
dsZ 1
�1
|x|p
3 + 2x
dx
Z 3⇡
�3⇡
x4 sen5 x dx
u v [a, b]
(a, b)
(uv)0(x) = u0(x)v(x) + u(x)v0(x),
[a, b] Z
u(x)v0(x)dx = u(x)v(x)�
Z
v(x)u0(x)dx.
du = u0(x)dx dv = v0(x)dx
Z
udv = uv �
Z
vdu.
Z b
a
udv = uv
���b
a
�
Z b
a
vdu.
h
h(x) = f(x)g(x) u = f(x) dv = g(x)dx
u = f(x) ; du = f 0(x)dx,
v =? ; dv = g(x)dx.
v
Z
g(x)dx
h = fg g
v Z
h(x)dx = uv �
Z
vdu
Z
h(x)dxZ
vdu
•
Z
vduZ
h(x)dx
•
Z
vduZ
h(x)dx
•
Z 1
0
xexdx
u = x ; du = dx,
v = ex ; dv = exdx.
Z 1
0
xexdx = xex
���1
0
�
Z 1
0
exdx.
ex
exZ 1
0
xexdx = xex
���1
0
� ex
���1
0
= e� (e� 1) = 1 .
⇤
Z
ex sen x dx
u = senx ; du = cos xdx,
v = ex ; dv = exdx.
Z
ex sen x dx = ex sen x�
Z
ex cosx dx.
Z
ex cosx
Z
ex cosx dx
ex cosx
u = cos x ; du = senxdx,
v = ex ; dv = exdx.
Z
ex cosx dx = ex cosx�
Z
ex(� sen x) dx.
Z
ex sen x dx = ex sen x� ex cosx�
Z
ex(sen x) dx.
2Z
ex sen x dx =
1
2
[ex sen x� ex cosx] + C.
⇤
Z
lnx dx
f
u = f(x) ; du = f 0(x)dx,
v = x ; dv = dx.
u = ln x ; du =
1
x
dx,
v = x ; dv = dx.
Z
lnx dx = x lnx�
Z
x
1
x
dx.
Z
lnx dx = x lnx� x+ C.
⇤
Z
sec3 x dx
u = secx ; du = secx tg xdx,
v = tg x ; dv = sec2 dx.
Z
sec3 x dx = secx tg x�
Z
tg2 x sec x dx.
tg2 x = sec2 x� 1
Z
sec3 x dx = secx tg x�
Z
sec3 x dx+
Z
sec x dx.
sec3 xZ
sec x dx = ln | sec x+ tg x|
Z
sec3 x dx =
1
2
[sec x tg x+ ln | sec x+ tg x|] + C.
⇤
u = sec2 x ; du = 2 sec2 x tg xdx,
v = ln | sec x+ tg x| ; dv = sec dx.
Z
ln | sec x+ tg x| sec2 x tg x dx,
Z
arctg x dx
Z
(lnx)3 dxZ
cossec3 x dx
Z
sen(ln x) dxZ
x sec2 x dx
Z
x3x dxZ
tg2 x sec3 x dx
Z ⇡
2
0
p
cosx sen3 x dx
cos2 x+ sen2 x = 1
tg2 x+ 1 = sec2 x
cos2 x =
1 + cos 2x
2
sen2 x =
1� cos 2x
2
senmx cosnx =
sen(m� n)x+ sen(n+m)x
2
Z
cos3 x dx
cos3 x = cos2 x cosx = (1�sen2 x) cos x
u = senx du = cos xdxZ
cos3 x dx =
Z
cos2 x cosx dx =
Z
(1� sen2 x) cos x dx
=
Z
(1� u2) du = u� u
3
3
+ C = senx� sen
3 x
3
+ C .
⇤
Z
cos3 x sen2 x dx
du
cos3 x sen2 x = cos2 x sen2 x cosx = (1 � sen2 x) sen2 x cosx
u = senxZ
cos3 x sen2 x dx =
Z
(1� sen2 x) sen2 x cosx dx =
Z
(1� u2)u2 du
=
Z
u2 � u4 du = u
3
3
� u
5
5
+ C =
sen3 x
3
� sen
5 x
5
+ C .
⇤
Z ⇡
0
sen2 x dx
Z ⇡
0
sen2 x dx =
Z ⇡
0
1� cos 2x
2
dx =
1
2
Z ⇡
0
1� cos 2x dx .
u = 2x
1
2
Z ⇡
0
1� cos 2x dx = 1
2
Z 2⇡
0
1� cosu du
2
=
1
4
(u� senu)
���2⇡
0
=
⇡
2
.
⇤
Z
cos4 x dx
Z
cos4 x dx =
Z ✓
1 + cos 2x
2
◆2
dx =
1
4
Z
1 + 2 cos 2x+ cos2 2x dx
=
x
4
+
1
2
Z
cos 2x dx+
1
4
Z
cos2 2x dx .
u = 2x
1
2
Z
cos 2x dx =
1
4
Z
cosu du =
1
4
senu+ C =
1
4
sen 2x+ C .
2x
x
1
4
Z
cos2 2x dx =
1
4
Z
1 + cos 4x
2
dx =
x
8
+
1
8
Z
cos 4x dx
=
x
8
+
1
32
Z
cosu du =
x
8
+
1
32
senu+ C =
x
8
+
1
32
sen 4x+ C .
u = 4x
du = 4dx Z
cos4 x dx =
3x
8
+
1
4
sen 2x+
1
32
sen 4x+ C .
⇤
Z
sen6 x dx
Z
sen6 x dx =
Z
(sen2 x)3 dx =
Z ✓
1� cos 2x
2
◆3
dx
=
1
8
Z
1� 3 cos 2x+ 3 cos2 2x� cos3 2x dx .
u = 2x du = 2dx
Z
sen6 x dx =
1
16
Z
1� 3 cosu+ 3 cos2 u� cos3 u du
=
u
16
+
3u
32
+
3 senu
64
� 1
16
senu+
1
48
sen3 u+ C
=
5x
16
� 1
64
sen 2x+
1
48
sen3 2x+ C .
⇤
Z
cos2 x sen4 x dx
Z
cos2 x sen4 x dx =
Z
sen4 x� sen6 x dx .
Z
cos2 x sen4 x dx =
3x
8
� 1
4
sen 2x+
1
32
sen 4x� 5x
16
+
1
64
sen 2x� 1
48
sen3 2x+ C
=
x
16
� 15
64
sen 2x+
1
32
sen 4x� 1
48
sen3 2x+ C .
⇤
Z
sen 3x cos 7x dx
m = 3 n = 7Z
sen 3x cos 7x dx=
1
2
Z
sen�4x+ sen 10x dx = cos 4x
8
� cos 10x
20
+ C .
⇤
Z ⇡/4
�⇡/4
sec4 x dx
u = tg x du = sec2 xdxZ ⇡/4
�⇡/4
sec4 x dx =
Z ⇡/4
�⇡/4
sec2 x sec2 x dx =
Z ⇡/4
�⇡/4
(1 + tg2 x) sec2 x dx
=
Z 1
�1
(1 + u2) du = (u+
u3
3
)
���1
�1
=
8
3
.
⇤
Z
tg3 x dx
u = tg x du = sec2 xZ
tg3 x dx =
Z
tg x(sec2 x� 1) dx =
Z
u du�
Z
tg x dx
=
u2
2
+ ln | cosx|+ C = tg
2 x
2
+ ln | cosx|+ ,
⇤
Z
tg4 x dx
Z
cos5 x dxZ
cos6 x dx
Z
sen x tg2 x dxZ
cos ⇡x sen x dx
Z
sec2n x dx n = 3, 4R
tg x sec3 x dx
R
cos2 x sen2 x dxR
sen x cos3 x dx
u = g(x)
h : I ! J C1(I)
C1(J)
x = h(✓)
Z
f(x)dx =
Z
f(h(✓))h0(✓)d✓.
G
f(h(✓))h0(✓) G(h�1(x)) f
G0(h�1(x))[h�1]0(x) = f(h(h�1(x)))h0(h�1(x))[h�1]0(x) = f(x).
p
a2 � x2 px2 � a2 pa2 + x2
cos2 ✓ = 1 � sen2 ✓ tg2 ✓ = sec2 ✓ � 1 sec2 ✓ = 1 + tg2 ✓
a > 0
(
p
a2 � x2) h(✓) = a sen ✓ [�⇡/2, ⇡/2]
h�1(x) = arcsen(x/a) x 2 [�a, a]
x = a sen ✓ ; dx = a cos ✓ d✓.
cos2 ✓ = 1� sen2 ✓
cos ✓ =
p
a2�x2
a
✓ tg ✓ cotg ✓
x
sen ✓ = xa .
x
a
p
a2 � x2
cos ✓ = CAH tg ✓ =
CO
CA cos ✓ =
p
a2�x2
a
tg ✓ = xp
a2�x2 .
cotg ✓ = 1tg ✓ sec ✓ =
1
cos ✓ cossec ✓ =
1
sen ✓
Z
dx
x
p
5� x2
x =
p
5 sen ✓ ; dx =
p
5 cos ✓ d✓,
Z
dx
x
p
5� x2 =
Z
(
p
5 cos ✓)d✓p
5 sen ✓(
p
5 cos ✓)
=
p
5
5
Z
cossec ✓ d✓,
Z
dx
x
p
5� x2 =
p
5
5
ln | cossec ✓ � cotg ✓|+ C.
x
sen ✓ =
xp
5
, cossec ✓ =
p
5
x
, cotg ✓ =
p
5� x2
x
.
Z
dx
x
p
5� x2 =
p
5
5
ln
���p5�p5� x2���� p5
5
ln |x|+ C.
⇤
Z
x2 dxp
25� x2
x = 5 sen ✓ ; dx = 5 cos ✓ d✓,
Z
x2 dxp
25� x2 =
Z
(5 sen ✓)2(5 cos ✓) d✓
(5 cos ✓)
= 25
Z
sen2 ✓ d✓.
Z
x2 dxp
25� x2 = 25{✓/2� sen(2✓)/4}+ C.
x
✓ = arcsen(x/5)
sen(2✓) = 2 sen ✓ cos ✓ = 2(x/5)(
p
1� (x/5)2).
Z
x2 dxp
25� x2 =
25
2
arcsen(x/5)� x
2
p
25� x2 + C.
⇤
(
p
a2 + x2) h(✓) = a tg ✓ (�⇡/2, ⇡/2)
h�1(x) = arctg(x/a) x 2 R
x = a tg ✓ ; dx = a sec2 ✓ d✓.
tg2 ✓ = sec2 ✓ � 1
sec ✓ =
p
a2+x2
a
tg ✓ = xa .
x
a
p
a2 � x2
sen ✓ = COH cos ✓ =
CA
H sen ✓ =
xp
a2+x2
cos ✓ = ap
a2+x2
.
Z
dxp
x2 + 16
x = 4 tg ✓ ; dx = 4 sec2 ✓ d✓,
Z
dxp
x2 + 16
=
Z
4 sec2 ✓ d✓
4 sec ✓
d✓ =
Z
sec ✓ d✓.
Z
dxp
x2 + 16
= ln | sec ✓ + tg ✓|+ C.
x
tg ✓ = x/4 sec ✓ =
p
16 + x2
4
Z
dxp
x2 + 16
= ln
��p16 + x2 + x��+ C.
⇤
Z
1
(x2 + 3)2
dx
x =
p
3 tg ✓ ; dx =
p
3 sec2 ✓ d✓,
Z
1
(x2 + 3)2
dx =
Z p
3 sec2 ✓
(3 tg2 ✓ + 3)2
d✓ =
p
3
9
Z
cos2 ✓ d✓
Z
1
(x2 + 3)2
dx =
p
3
9

✓
2
+
sen(2✓)
4
�
+ C.
x
✓ = arctg
xp
3
sen ✓ =
xp
x2 + 3
cos ✓ =
p
3p
x2 + 3
.
Z
1
(x2 + 3)2
dx =
p
3
18
h
arctg
xp
3
+
p
3x
x2 + 3
i
+ C.
⇤
(
p
x2 � a2) h(✓) = a sec ✓
[0, ⇡/2) (⇡/2, ⇡] h�1(x) = arcsec(x/a)
x 2 [a,+1) x 2 (�1, a]
x = a sec ✓ ; dx = a sec ✓ tg ✓ d✓,
0  ✓ < ⇡/2 ⇡/2 < ✓  ⇡
tg2 ✓ = sec2 ✓ � 1
tg ✓ =
p
x2�a2
a tg ✓ = �
p
x2�a2
a
sen ✓ =
p
x2�a2
x cos ✓ =
a
x .
Z
x2 dxp
x2 � 9 x 2 (3,+1)
sec ✓ = xa .
x
a
p
x2 � a2
x = 3 sec ✓ ; dx = 3 sec ✓ tg ✓ d✓,
Z
x2 dxp
x2 � 9 =
Z
(3 sec ✓)2(3 sec ✓ tg ✓) d✓
(3 tg ✓)
= 9
Z
sec3 ✓ d✓.
Z
sec3 ✓ d✓ =
1
2
sec ✓ tg ✓ +
1
2
ln | sec ✓ + tg ✓|+ C.
Z
x2 dxp
x2 � 9 =
9
2
sec ✓ tg ✓ +
9
2
ln | sec ✓ + tg ✓|+ C.
x
sec ✓ =
x
3
, tg ✓ =
p
x2 � 9
3
.
Z
x2 dxp
x2 � 9 =
x
p
x2 � 9
2
+
9
2
ln
��x+px2 � 9��+ C.
⇤
Z 8
4
p
x2 � 16
x2
dx
x = 4 sec ✓ ; dx = 4 sec ✓ tg ✓ d✓,
Z 8
4
p
x2 � 16
x2
dx =
Z ⇡
3
0
(4 tg ✓)(4 sec ✓ tg ✓) d✓
(4 sec ✓)2
=
Z ⇡
3
0
[sec ✓ � cos ✓] d✓,
Z 8
4
p
x2 � 16
x2
dx = [ln | sec ✓ + tg ✓|� sen ✓]
���⇡3
0
.
Z 8
4
p
x2 � 16
x2
dx = [ln(2 +
p
3)�
p
3
2
]� [ln 1� 0] = ln(2 +p3)�
p
3
2
.
⇤
Z
x3p
1� x2 dx
Z 2
�2
1
4x2 + 9
dxZ p
8� 2x� x2 dx
Z
1
x2
p
1 + x2
dxZ p
x2 + 2x dx
Z a
0
x2
p
a2 � x2 dx
x2
a2 +
y2
b2 = 1
9x2 � 4y2 = 36 x = 3
Z
1
x2 + 1
dx = arctg x + CZ
1
x2 � 1 dx
1
x2 � 1 =
1/2
x� 1 �
1/2
x+ 1
,
Z
1
x2 � 1 dx =
Z
1/2
x� 1 dx�
Z
1/2
x+ 1
dx =
1
2
ln |x� 1|� 1
2
ln |x+ 1|+ C .
f(x) = p(x)/q(x) q(x)
p(x) f
q(x)
i) (ax+ b)m m
�b/a
ii) (ax2 + bx+ c)k
k
(ax + b)m
A1
ax+ b
+
A2
(ax+ b)2
+ ...+
Am
(ax+ b)m
,
A1, A2, · · · , Am
(ax2 + bx+ c)k
B1x+ C1
ax2 + bx+ c
+
B2x+ C2
(ax2 + bx+ c)2
+ ...+
Bkx+ Ck
(ax2 + bx+ c)k
.
p(x)/q(x)
A1, A2, ..., B1, .., C1, ...
A,B,C,D...
Z
x+ 2
x(x� 1)(2x+ 1) dx .
q(x)
x+ 2
x(x� 1)(2x+ 1) =
A
x
+
B
x� 1 +
C
2x+ 1
,8x 6= 0, 1,�1/2 .
x+ 2 = A(x� 1)(2x+ 1) +Bx(2x+ 1) + Cx(x� 1),8x 6= 0, 1,�1/2 .
R x = 0, 1,�1/2 x
x = 0 2 = �A
x = 1 3 = 3B x = �1/2 3/2 = 3C
4
A = �2
B = 1 C = 2
x+ 2
x(x� 1)(2x+ 1) =
�2
x
+
1
x� 1 +
2
2x+ 1
Z
x+ 2
x(x� 1)(2x+ 1) dx =
Z �2
x
dx+
Z
1
x� 1 dx+
Z
2
2x+ 1
dx
= �2 ln |x|+ ln |x� 1|+ ln |2x+ 1|+ C .
⇤
Z
5x2 + 20x+ 6
x3 + 4x2 + 4x
dx
q(x) = x3+4x2+4x = x(x2+4x+4) = x(x+2)2
5x2 + 20x+ 6
x3 + 4x2 + 4x
=
A
x
+
B
x+ 2
+
C
(x+ 2)2
,8x 6= 0,�2 .
5x2 + 20x + 6 = A(x + 2)2 + Bx(x + 2) + Cx
x = 0 x = �2
A = 3/2 C = 7
x
A C x = 1 B = 7/2
5x2 + 20x+ 6
x3 + 4x2 + 4x
=
3/2
x
+
7/2
x+ 2
+
7
(x+ 2)2
;
Z
5x2 + 20x+ 6
x3 + 4x2 + 4x
dx =
3
2
Z
1
x
dx+
7
2
Z
1
x+ 2
dx+ 7
Z
1
(x+ 2)2
dx
=
3
2
ln |x|+ 7
2
ln |x+ 2|� 7
x+ 2
+ C .
⇤
Z
1
x(x2 + 4)
dx
q(x)
x = 0
1
x(x2 + 4)
=
A
x
+
Bx+ C
x2 + 4
,8x 6= 0.
1 = A(x2 +4)+ (Bx+C)x = (A+B)x2 +Cx+4A
A + B = 0 C = 0 4A = 1 A = 1/4 B = �1/4 C = 0
Z
1
x(x2 + 4)
=
1
4
Z
1
x
dx� 1
4
Z
x
x2 + 4
dx =
1
4
ln |x|� 1
8
ln(x2 + 4) + C ,
u = x2+4 du = 2xdx
⇤
Z
x� 2
x(x4 � 8x2 � 9)
q(x)
x4 � 8x2 � 9 = 0 t = x2
t t2 � 8t � 9 = 0 t = 9 t = �1
t2 � 8t � 9 = (t � 9)(t + 1)
x4 � 8x2 � 9 = (x2 � 9)(x2 + 1) = (x� 3)(x+ 3)(x2 + 1)
x� 2
x(x4 � 8x2 � 9) =
x� 2
x(x� 3)(x+ 3)(x2 + 1) =
A
x
+
B
x� 3 +
C
x+ 3
+
Dx+ E
x2 + 1
.
x� 2 = A(x� 3)(x+ 3)(x2 + 1) +Bx(x+ 3)(x2 + 1) + Cx(x� 3)(x2 + 1)
+ (Dx+ E)x(x� 3)(x+ 3) .
x x = �3 x = 0 x = 3
C = �1/36 A = 2/9 B = 1/180
x = 1 D + E = �3/10 x = �1 E � D = 13/30
E = 1/15 D = �11/30Z
x� 2
x(x� 3)(x+ 3)(x2 + 1) dx =
2
9
Z
1
x
dx+
1
180
Z
1
x� 3 dx�
1
36
Z
1
x+ 3
dx
� 11
30
Z
x
x2 + 1
dx+
1
15
Z
1
x2 + 1
dx
=
2
9
ln |x|+ 1
180
ln |x� 3|� 1
36
ln |x+ 3|
� 11
60
ln(x2 + 1) +
1
15
arctg x+ C .
⇤
Z
e3x + 1
(e2x + 4)2
dx
u = ex du = exdxZ
e3x + 1
(e2x + 4)2
dx =
Z
u3 + 1
u(u2 + 4)2
du q(u)
u = 0
u3 + 1
u(u2 + 4)2
=
A
u
+
Bu+ C
u2 + 4
+
Du+ E
(u2 + 4)2
,8u 6= 0.
1 + u3 = A(u2 + 4)2 + (Bu+ C)(u2 + 4)u+ (Du+ E)u
= A(u4 + 8u2 + 16) +Bu4 + 4Bu2 + Cu3 + 4Cu+Du2 + Eu.
A+B = 0, C = 1, 8A+ 4B +D = 0, 4C + E = 0, 16A = 1 .
A = 1/16, B = �1/16, C = 1, D = �1/4, E = �4
1 + u3
u(u2 + 4)2
=
1
16u
� 1
16
u
u2 + 4
+
1
u2 + 4
� 1
4
u
(u2 + 4)2
� 4
(u2 + 4)2
.
v = u2 + 4 dv = 2udu
u = 2 tg ✓ du = 2 sec2 ✓d✓Z
u3 + 1
u(u2 + 4)2
du =
1
16
ln |u|� 1
32
ln(u2 + 4) +
1
2
arctg(u/2) +
1
8(u2 + 4)
�
1
4
arctg(u/2)�u
2(u2 + 4)
+ C
x
Z
e3x + 1
(e2x + 4)2
dx =
x
16
� 1
32
ln(e2x+4)+
1
4
arctg(ex/2)+
1
8(e2x + 4)
� e
x
2(e2x + 4)
+C.
⇤
f(x) = p(x)/q(x)
q(x) p(x)
f(x) = g(x) + r(x)/q(x),
g(x) r(x)/q(x)
r(x) q(x)
r(x)/q(x)
Z
x3 � 2x2 + 1
x2 � 5x+ 6 dx
x3 � 2x2 + 1
x2 � 5x+ 6 = x+ 3 +
9x� 17
x2 � 5x+ 6
9x� 17
x2 � 5x+ 6 =
9x� 17
(x� 2)(x� 3) =
A
x� 2 +
B
x� 3;
9x � 17 = A(x � 3) + B(x � 2) x = 2
x = 3 A = �1 B = 10
9x� 17
x2 � 5x+ 6 =
�1
x� 2 +
10
x� 3 .
Z
x3 � 2x2 + 1
x2 � 5x+ 6 =
Z
x+ 3 dx+
Z �1
x� 2 dx+
Z
10
x� 3 dx
=
x2
2
+ 3x� ln |x� 2|+ 10 ln |x� 3|+ C .
⇤
Z
x2
x2 + 1
dx
x2 x2+1
x2
x2 + 1
= 1� 1
x2 + 1
Z
x2
x2 + 1
dx =
Z
1 dx�
Z
1
x2 + 1
dx = x� arctg x+ C.
⇤
•
• q(x)
•
Z
dx
x2 � 5x+ 4
Z
x+ 3
2x3 � 8x dxZ
x3
x2 � 2x+ 1 dx
Z
dx
(x2 � 1)2Z
s4 + 81
s(s2 + 9)2
ds
Z
y4 + y2 � 1
y3 + y
dyZ
sen x
cos2 x+ cosx� 2 dx
Z
x4 arctg x dxZ
x3 + 1
(x+ 1)3
dx
Z
dx
x3 + 1
x
xm(a� bxn)p
x
x
Z
x
1
2
1 + x
3
2
dx
x
p1
q1 , · · · , x
pj
qj
x = tk ; dx = ktk�1dt,
k q1, · · · , qj
Z
x1/2 dx
x3/4 + 1
4 {2, 4}
x = t4 ; dx = 4t3dt .
Z
x1/2 dx
x3/4 + 1
=
Z
t2(4t3) dt
t3 + 1
= 4
Z h
t2 � t
2
t3 + 1
i
dt,
Z
x1/2 dx
x3/4 + 1
=
4
3
t3 � 4
3
ln |t3 + 1|+ C.
x t = x1/4
Z
x1/2 dx
x3/4 + 1
=
4
3
x3/4 � 4
3
ln(x3/4 + 1) + C.
⇤
Z p
x+ 4
x
dx
u = x + 4 du = dx
Z p
x+ 4
x
dx =
Z p
u
u� 4 du.
u = t2 ; du = 2tdt
Z p
u
u� 4 du =
Z
t(2t) dt
t2 � 4 = 2
h Z
dt+
Z
1
t� 2 dt�
Z
1
t+ 2
dt
i
,
Z p
x+ 4
x
dx = 2t+ ln |t� 2
t+ 2
|+ C.
x t =
p
x+ 4Z p
x+ 4
x
dx = 2
p
x+ 4 + 2 ln
���px+ 4� 2p
x+ 4 + 2
���+ C.
⇤
z = tg(x/2) ; dz =
1
2
sec2(x/2)dx
sen x cosxZ
sen x
1 + cos x
dx z
dx =
2 dz
1 + z2
sen x =
2z
1 + z2
cosx =
1� z2
1 + z2
.
sen x = 2 sen(x/2) cos(x/2) =
2 tg(x/2)
sec2(x/2)
=
2 tg(x/2)
1 + tg2(x/2)
,
cosx = 2 cos2(x/2)� 1 = 2
sec2(x/2)
� 1 = 1� tg
2(x/2)
1 + tg2(x/2)
.
Z
sen x
1 + cos x
dx
Z
sen x
1 + cos x
dx =
Z 2z
1+z2
1 + 1�z21+z2
2
1 + z2
dz =
Z
2z
1 + z2
dz,
Z
sen x
1 + cos x
dx = ln(1 + z2) + C.
x
Z
sen x
1 + cos x
dx = ln(sec2(x/2)) + C.
⇤
sen(x/2) =
p
2 senx
2
p
1 + cos x
cos(x/2) =
r
1 + cosx
2
.
Z
dx
4� 5 senx
Z
dx
4� 5 senx =
Z
1
[4� 5( 2z1+z2 )]
2 dz
1 + z2
=
Z
dz
2z2 � 5z + 2
Z
dz
2z2 � 5z + 2 = 1/3
Z
dz
z � 2 � 1/3
Z
dz
z � 1/2
Z
dx
4� 5 senx = 1/3 ln |z � 2|� 1/3 ln |z � 1/2|+ C.
xZ
dx
4� 5 senx =
1
3
ln
��� tg(x/2)� 2
tg(x/2)� 1/2
���+ C.
⇤
xm(a� bxn)p
xm(a� bxn)p a b
m, n p
p = �1Z
x�2/3(1 + x2/3)�1 dx
z = x
2
3
3
2
Z
z�
1
2 (1 + z)�1dz.
z = t2
3
2
Z
t�1(1 + t2)�1(2t)dt = 3
Z
(1 + t2)�1dt = 3arctg t+ C.
Z
x�2/3(1 + x2/3)�1 dx = 3arctg(
p
z) + C = 3arctg(x
1
3 ) + C.
⇤
p+ m+1n = �2Z
1
x2(1 + x2)3/2
dx
z = x2
1
2
Z
z�
3
2 (1 + z)�
3
2dz =
1
2
Z
z�3(
1 + z
z
)�
3
2dz .
t = (1+zz )
1
2
�
Z
t2 � 1
t2
= �t� 1
t
+ C.
Z
x1/3(2 + x2/3)1/4 dx = �
p
1 + x2
x
� xp
1 + x2
+ C
⇤
m+1
n = 2Z
(1 + x1/2)3/4 dx
t = 1 + x
1
2
Z
(1 + x1/2)3/4 dx = 2
Z
t3/4(t� 1) dt.
Z
(1 + x1/2)3/4 dx =
8
11
(1 + x
1
2 )11/4 � 8
7
(1 + x
1
2 )7/4 + C .
⇤
xm(a� bxn)p a b
m, n p
p, m+1n p+
(m+1)
n
Z
1
1 + senx
dx
Z
1
1 + cos x
dxZ
1
cosx+ sen x
dx
Z p
x
4
p
x3 + 1
dxZ p
x3 � 3px
4
p
x
dx
Z
1
4
p
x+
p
x
dxZ
(1 + x2)�
3
2 dx
Z
x5(1 + x3)
2
3 dxZ
x�2/3(1 + x1/3)
1
2 dx
x R x
f [a, b]
z
x
y
x
P = ({t0, t1, · · · , tn�1, tn}, {t⇤1, · · · , t⇤n�1, t⇤n})
n
[t0, t1], [t1, t2], · · · , [tn�1, b] {f(t⇤1), f(t⇤2), · · · , t⇤n}
{a = t0 < t⇤1 < t1 < · · · < tn�1 < t⇤n < tn = b}.
R
x
f(ti), i = 0, .., n� 1. ti = (ti)⇤
y
x
z
R x
r = f(t⇤i ) �ti = ti� ti�1
Vi = ⇡(f(t⇤i ))
2�ti
V (S) = lim
||P||!0
nX
i=1
⇡[f(t⇤i )]
2�ti,
⇡f 2
S R
f : [a, b]! R x
V (S) = ⇡
Z b
a
[f(x)]2dx,
• f
• x
r(x) = |f(x)|
V (S) = ⇡
Z b
a
[r(x)]2dx.
R
y =
p
x x x 2 [0, 2]
R
r(x) =
p
x
V = ⇡
Z 2
0
(
p
x)2 dx = ⇡
Z 2
0
x dx = ⇡
x2
2
���2
0
= 2⇡ u.v ,
u.v ⇤
R
y = sen 2x x x 2 [0, ⇡]
R r(x) = | sen 2x|
V = ⇡
Z ⇡
0
sen2(2x) dx = ⇡
Z ⇡
0
1� cos 4x
2
dx =
⇡
2
[x� sen 4x
4
]
���⇡
0
=
⇡2
2
u.v .
⇤
R y = f(x) x 2 [a, b]
y = c
R
f y = f(x) � c x 2 [a, b] x
r(x) = |f(x)� c|
V = ⇡
Z b
a
(f(x)� c)2 dx,
R
x
R
y = ex x 2 [0, 1] y = 1 y = 1
R
r(x) = ex � 1
V = ⇡
Z 1
0
(ex � 1)2 dx = ⇡
Z 1
0
e2x � 2ex + 1 dx
= ⇡[
e2x
2
� 2ex + x]
���1
0
=
⇡
2
[e2 � 4e+ 5]u.v.
⇤
R
y = tg x x 2 [�⇡
4
,
⇡
4
] y = �1
y = �1
R
r(x) = tg x+ 1
V = ⇡
Z ⇡
4
�⇡4
(tg x+ 1)2 dx = ⇡
Z ⇡
4
�⇡4
tg2 x+ 2 tg x+ 1 dx .
tg2 x = sec2 x� 1
V = ⇡
Z ⇡
4
�⇡4
sec2 x+ 2 tg x dx = ⇡[tg x+ 2 ln | sec x|
���⇡4
�⇡4
= 2⇡ u.v.
⇤
0
<
R
-π/4 π/4
-1
1
y=-1
f g [a, b] f(x)  g(x) 8x 2 [a, b]
y = c
R y = c
x
y
z
y = g(x) y = c R(x) = g(x) � c
y = f(x)
r(x) = f(x)� c
R(x)
r(x)
V = ⇡
Z b
a
(g(x)� c)2 � (f(x)� c)2 dx = ⇡
Z b
a
R2(x)� r2(x) dx .
R
y = 4� x2 y = 2� x y = �2
4� x2 = 2� x, x = 2 ou x = �1 .
R(x) = 4� x2 � (�2) = 6� x2 r(x) = 2� x� (�2) = 4� x
V = ⇡
Z 2
�1
(6� x2)2 � (4� x)2 dx = ⇡
Z 2
�1
36� 12x2 + x4 � 16 + 8x� x2dx
= ⇡
Z 2
�1
20� 13x2 + x4 + 8xdx = [20x� 13x
3
3
+
x5
5
+ 4x2]
���2
�1
=
198⇡
5
⇤
x = c
y
x = g(y) x = f(y) y 2 [a, b] x = c
V = ⇡
Z b
a
(g(y)� c)2 � (f(y)� c)2 dy = ⇡
Z b
a
R2(y)� r2(y) dy
y x = c
R
y =
p
x y = x x = 1
y =
p
x x = y2
V = ⇡
Z 1
0
(y2 � 1)2 � (y � 1)2 dy = ⇡
Z 1
0
y4 � 2y2 + 1� y2 + 2y � 1 dy
= ⇡
Z 1
0
y4 � 3y2 + 2y dy = ⇡[y
5
5
� y3 + y2]
���1
0
=
⇡
5
⇤
R
y = 4� x2 y = 2� x x = 2
R1 R2
R1 R1 = 2�(�p4� y)
y 2 [3, 4] r1 = 2�p4� y y 2 [3, 4] R2
R2 = 2 � (2 � y) r2 = r1 = 2 � p4� y y 2 [0, 3]
V = V1 + V2
V1 = ⇡
Z 4
3
[2� (�p4� y)2 � (2�p4� y)2 dy
= ⇡
Z 4
3
8
p
4� y dy = 16⇡
3
V2 = ⇡
Z 3
0
[2� (2� y)]2 � (2�p4� y)2 dy
= ⇡
Z 3
0
8 + 9y + y2 + 4
p
4� y dy = 553⇡
6
⇤
y = x y = 5x� x2 x = 4
y = x y = 5x� x2 y = 0
x = y2 � 4y x = 2y � y2 x = �4
y =
p
25� x2 y = 0 x = 2 x = 4 y = 0
y = e�x x = 2 y = 1 y = 2
y = ln x y = 1 y = 2 x = 0 y
r > 0 V =
4
3
⇡r3
V = ⇡
Z ⇡
�⇡
(1� cosx)2dx
z
x
y
xy y
z
x
y
R r
h
V = ⇡R2h� ⇡r2h = ⇡h(R + r)(R� r)
R f g
[a, b] g(x)  f(x) 8x 2 [a, b] x = c
x = c
a = x0 < x1 < x2 < ... < xn�1 < xn = b [a, b]
h = g(ci)� f(ci) ci = xi�1 + �xi
2
ci = xi� �xi
2
ci [xi�1, xi]
R = xi � c = ci + �xi
2
� c
r = xi�1 � c = ci � �xi
2
� c h = g(ci)� f(ci)
V (S) '
nX
i=1
2⇡[g(ci)� f(ci)](ci � c)�xi
f g
[a, b]
V (S) = lim
||P||!0
nX
i=1
2⇡(ci � c)[g(ci)� f(ci)]�xi .
V (S) = 2⇡
Z b
a
(x� c)[g(x)� f(x)] dx
x�c c�x
V (S) = 2⇡
Z b
a
|x� c|[g(x)� f(x)] dx
•
V (S) = 2⇡
Z b
a
( )( ) dx .
• g � 0 x
x 2 [a, b] y
V = 2⇡
Z b
a
|x|g(x) dx,
x = 0 f
• f(y)  g(y)
y 2 [a, b] y = c
x y
V (S) = 2⇡
Z b
a
|y � c|[g(y)� f(y)]dy.
y
R
y = 4 y = x2 y
h = 4� x2 r = x
V = 2⇡
Z 2
0
x(4� x2) dx = 2⇡(4x
2
2
� x
4
4
)
���2
0
= 8⇡.
⇤
x2� 6x+ y2 = 0 x = �1
(x�3)2+y2 = 9
h =
p
9� (x� 3)2 r = x+ 1
V = 2⇥ 2⇡
Z 6
0
(x+ 1)
p
9� (x� 3)2 dx .
x�3 = 3 sen ✓
V = 4⇡
Z ⇡/2
�⇡/2
(4 + 3 sen ✓)9 cos2 ✓ d✓
= 144⇡
Z ⇡/2
�⇡/2
cos2 ✓ d✓ + 108⇡
Z ⇡/2
�⇡/2
cos2 ✓ sen ✓ d✓
= 144⇡
Z ⇡/2
�⇡/2
1 + cos 2✓
2
d✓ � 108⇡
Z 0
0
u2 du
= 72⇡[✓ +
sen 2✓
2
]
���⇡/2
�⇡/2
� 0 = 72⇡2 .
⇤
R f(x) = x5+x2+x+2
x x 2 [0, 1]
R y
h = x5 + x2 + x+ 2 r = x
V = 2⇡
Z 1
0
x(x5 + x2 + x+ 2) dx = 2⇡(
x7
7
+
x4
4
+
x3
3
+ x2)
���2
0
=
1216⇡
21
.
f
y ⇤
R
R x
h = y2 � (3y2 � 2) r = y
V = 2⇡
Z 1
0
y[y2 � (3y2 � 2)] dy = 2⇡
Z 1
0
�2y3 + 2y dy = ⇡ .
R =
r
x+ 2
3
r =
p
x
V = ⇡
Z 1
�2
 r
x+ 2
3
!2
dx� ⇡
Z 1
0
�p
x
�2
dx
= ⇡
Z 1
�2
x+ 2
3
dx� ⇡
Z 1
0
x dx = ⇡
⇤
y = x y = 5x� x2 x = 4
y = x y = 5x� x2 y = 0
x = y2 � 4y x = 2y � y2 x = �4
y =
p
25� x2 y = 0 y = 0
y = e�x x = 2 y = 1 y = 2
y = ln x y = 1 y = 2 x = 0 x
y = ln x y = 1 y = 2 x = 0 y
r 2⇡r
20⇡ ⇡ 62, 8318
⇡
C
C
f : [a, b] ! R
P = {a = t0 < t1 < · · · < tn�1 < tn = b},
{(t, f(t))}t2P .
C
f : [a, b]! R
L(f, [a, b]) = lim
||P||!0
nX
i=1
p
(ti � ti�1)2 + (f(ti)� f(ti�1))2
[a, b] C
C f : [a, b]! R
f 0 : [a, b]! R
L(f, [a, b]) =
Z b
a
p
1 + [f 0(x)]2dx .
f(x) = ln(cos x) [0, ⇡/4]
f 0(x) = � tg x [0, ⇡/4]
L(f, [0, ⇡/4]) =
Z ⇡/4
0
p
1 + [� tg x]2dx =
Z ⇡/4
0
sec xdx.
Z
sec xdx = ln | sec x+ tg x|+ C,
L(f, [0, ⇡/4]) = ln �p2 + 1�.
⇤
f(x) = x2 � 2x + 5
(1, 4) (3/2, 17/4)
f 0(x) = 2x�2 [1, 3/2]
L(f, [1, 3/2]) =
Z 3/2
1
p
1 + [2x� 2]2dx .
u = 2x � 2 du = 2dx
1
2
Z 1
0
p
1 + u2du .
u = tg ✓ du = sec2 ✓d✓
Z ⇡/4
0
sec3 ✓d✓.
Z
sec3 x dx =
1
2
{sec x tg x+ ln | sec x+ tg x|}+ C ,
L(f, [1, 3/2]) = 1
4
[
p
2 + ln(
p
2 + 1)].
⇤
C f : [0, 2/⇡]! R
f(x) =
8<:x sen( 1x) x 6= 00 x = 0.
f
n � 1
[0, 2/⇡]
Pn = {0 < 2
n⇡
<
2
(n� 1)⇡ < · · · <
2
3⇡
<
2
2⇡
<
2
⇡
} .
Pn
2
⇡
{1 + 1
3
+
1
3
+
1
5
+
1
5
+ · · ·+ 1
n
+
1
n
},
f
2
⇡
{1
2
+
1
3
+
1
4
+
1
5
+ · · ·+ 1
n
+
1
n+ 1
},
n
C
⇤
Pn ||Pn||! 0 n!1
y = e
x
2 + e�
x
2 x = 0 x = 2
y = ln x x =
p
3 x =
p
8
y = 1� ln(cosx) x = 0 x = ⇡4
y = ln(1� x2) x = 0 x = 12
x = y
5
6 +
1
10y3 y = 1 y = 2
x = 13
p
y(y � 3) y = 1 y = 9
f : [a, b]! R
[a,+1) [�1, b) (�1,+1)
f : [a,+1) ! R
[a, b] a < b
Z 1
a
f(x)dx = lim
b!1
Z b
a
f(x)dx ,
f : (�1, b] ! R
[a, b] a < b
Z b
�1
f(x)dx = lim
a!�1
Z b
a
f(x)dx ,
f : [�1,+1)! R
[a, b] a < b
Z 1
�1
f(x)dx =
Z c
�1
f(x)dx+
Z 1
c
f(x)dx,
Z 1
�1
f(x)dx c 2 R
↵ 2 R
Z 1
1
1
x↵
dx
Z b
1
1
x↵
dx =
8<: b
1�↵�1
1�↵ , ↵ 6= 1,
ln b, ↵ = 1.
Z 1
1
1
x↵
dx = lim
b!1
Z b
1
1
x↵
dx =
8<: 1↵�1 , ↵ > 1,1, ↵  1.
↵ > 1Z 1
1
1
x↵
dx =
1
↵� 1 .
⇤
f(x) = 11+x2
f(x) = 1
1+x2
Z 1
�1
1
1 + x2
dx =
Z 0
�1
1
1 + x2
dx+
Z 1
0
1
1 + x2
dx.
Z b
a
1
1 + x2
dx = arctg(b)� arctg(a).
Z 0
�1
1
1 + x2
dx = lim
a!�1
{arctg(0)� arctg(a)} = �(�⇡
2
),
Z 1
0
1
1 + x2
dx = lim
b!1
{arctg(b)� arctg(0)} = ⇡
2
.
Z 1
�1
1
1 + x2
dx = ⇡.
⇤
Z 1
0
xe�x dx
Z b
0
xe�xdx = �e�x[x+ 1]
���b
0
= �e�b[b+ 1] + 1.
Z 1
0
xe�xdx = 1� lim
b!1
e�b[b+ 1].
f(x) = xe�x
Z 1
0
xe�xdx = 1.
⇤
Z 1
2
1
x2 � 1 dx
f(x) = 1
x2�1
Z b
2
1
x2 � 1dx =
1
2
nZ b
2
1
x� 1dx�
Z b
2
1
x+ 1
dx
o
=
1
2
ln
���x� 1
x+ 1
������b
2
=
1
2
ln
���b� 1
b+ 1
���� 1
2
ln
⇣1
3
⌘
=
1
2
ln
���b� 1
b+ 1
���+ ln 3
2
.
Z 1
2
1
x2 � 1 dx =
ln 3
2
+
1
2
lim
b!1
ln
���b� 1
b+ 1
���.
Z 1
2
1
x2 � 1 dx =
ln 3
2
.
⇤
Z 1
�1
1
x2 + 2x+ 2
dx
Z 1
0
x sen x dxZ 1
2
1
x5
dx
Z 1
�1
e�|x| dxZ 1
1
lnx
x
dx
Z 0
�1
1
(x� 8) 23 dxZ 1
0
e�ax sen(bx) dx a > 0
Z 1
0
e�ax cos(bx) dx a > 0
f : [a, b] ! R
f : [a, b)! R
[a, c] c < b
Z b
a
f(x)dx = lim
c!b�
Z c
a
f(x)dx,
f : (a, b]! R
[c, b] c < b
Z b
a
f(x)dx = lim
c!a+
Z b
c
f(x)dx,
f : [a, c)[(c, b]! R
[a, c) (c, b]
Z b
a
f(x)dx =
Z c
a
f(x)dx+
Z b
c
f(x)dx,
f(x) = 1x↵ (0, 1] ↵ > 0
↵ > 0
Z 1
0
1
x↵
dx
Z 1
c
1
x↵
dx =
8<:1�c
1�↵
1�↵ , ↵ 6= 1,
� ln c, ↵ = 1.
Z 1
0
1
x↵
dx = lim
c!0+
Z 1
c
1
x↵
dx =
8<: 11�↵ , 0 < ↵ < 1,1, ↵ � 1.
0 < ↵ < 1
Z 1
0
1
x↵
dx =
1
1� ↵ .
⇤
f(x) = 1
(1�x) 13
[0, 2]
x = 1
Z 2
0
1
(1� x) 13 dx
f(x) = 1
(1�x) 13
[0, 2]
[0, 1] [1, 2]
Z 2
0
1
(1� x) 13 dx =
Z 1
0
1
(1� x) 13 dx+
Z 2
1
1
(1� x) 13 dx.
u = 1� x du = �dx
Z b
0
1
(1� x) 13 dx =
Z 1
1�b
u�
1
3du =
3
2
[1� (1� b) 23 ],
Z 2
a
1
(1� x) 13 dx =
Z 1�a
�1
u�
1
3du =
3
2
[(1� a) 23 � (�1) 23 ]
0 < b < 1 1 < a < 2
Z 1
0
1
(1� x) 13 dx =
3
2
lim
b!1�
[1� (1� b) 23 ] = 3
2
,
Z 2
1
1
(1� x) 13 dx =
3
2
lim
a!1+
[(1� a) 23 � 1] = �3
2
.
Z 2
0
1
(1� x) 13 dx = 0.
⇤
f(x) = 1(1�x)2 [0, 2]
x = 1
Z 2
0
1
(1� x)2dx
f(x) = 1
(1�x)2
[0, 2]
[0, 1] [1, 2]
Z 2
0
1
(1� x)2dx =
Z 1
0
1
(1� x)2dx+
Z 2
1
1
(1� x)2dx,
u = 1� x du = �dx
Z b
0
1
(1� x)2dx =
Z 1
1�b
u�2du = (1� b)�1 � 1,
Z 2
a
1
(1� x)2dx =
Z 1�a
�1
u�2du = 1� (1� a)�1,
0 < b < 1 1 < a < 2
Z b
0
1
(1� x)2dx = limb!1�(1� b)
�1 � 1 = +1,
Z 2
a
1
(1� x)2dx = 1� lima!1+(1� a)
�1 = +1.
⇤
u = x � 1
Z 2
0
1
(1� x) 13 dx =
3
2
(x� 1)2/3
���2
0
=
3
2
[1� 1] = 0,
Z 2
0
1
(1� x)2dx = �(1� x)
�1
���2
0
= �[�1� 1] = 2,
f(x) = 1p
x(1�x) [0, 1]
x = 0 x = 1Z 1
0
1p
x(1� x)dx
Z 1
0
1p
x(1� x)dx =
Z 1/2
0
1p
x(1� x)dx+
Z 1
1/2
1p
x(1� x)dx,
f(x) = 1p
x(1�x)
u =
p
x du = 12pxdx
Z 1/2
a
1p
x(1� x)dx = 2arcsen
p
x
���1/2
a
=
⇡
3
� 2 arcsenpa,
Z b
1/2
1p
x(1� x)dx = 2arcsen
p
x
���b
1/2
= 2arcsen
p
b� ⇡
3
0 < a < 1/2 1/2 < b < 1
Z 1/2
0
1p
x(1� x)dx =
⇡
3
� 2 lim
a!0+
arcsen
p
a =
⇡
3
,
Z 1
1/2
1p
x(1� x)dx = 2 limb!1� arcsen
p
b� ⇡
3
= 2
⇡
2
� ⇡
3
.
Z 1
0
1p
x(1� x)dx = ⇡.
⇤
Z 1
�1
1
x4
dx
Z 1
0
1p
1� x2 dxZ 1
0
lnx dx
Z ⇡
2
0
cosxp
1� sen x dxZ 3
1
x2p
x3 � 1 dx
Z 1
1
1
x
p
x2 � 1 dxZ 5
1
1
(x� 3)4 dx
f : [a,1)! R g : [a,1)! R
[a, t] t > a f(x) � g(x) � 0 x � a
Z 1
a
f(x)dx
Z 1
a
g(x)dx
Z 1
a
g(x)dx
Z 1
a
f(x)dx
f
g f x
g x
f g
f : [a,1)! R g : [a,1)! R
[a, t] t > a f(x) � 0 g(x) > 0 x � a
� > 0
lim
x!1
f(x)
g(x)
= �,
Z 1
a
f(x)dx
Z 1
a
g(x)dx
x f �g
f g
g
f
f : [a, b) ! R g : [a, b) ! R
[a, c] c < b
f(x) � g(x) � 0 x 2 [a, b)Z b
a
f(x)dx
Z b
a
g(x)dx
Z b
a
g(x)dx
Z b
a
f(x)dx
f : [a, b) ! R g : [a, b) ! R
[a, c] c < b
f(x) � 0 g(x) > 0 x 2 [a, b) � > 0
lim
x!b�
f(x)
g(x)
= �,
Z b
a
f(x)dx
Z b
a
g(x)dx
R1
a f(x) dxR1
a|f(x)| dxR1
a f(x) dx
��� Z 1
a
f(x) dx
���  Z 1
a
|f(x)| dx.
Z 1
1
e�x
2
dx
Z b
1
e�x
2
dx
0  e�x2  e�x x � 1
Z b
1
e�xdx = 1� e�b
b > 1
Z 1
1
e�xdx = lim
b!1
1� e�b = 1.
Z 1
1
e�x
2
dx ⇤
Z 1
1
sen x e�x
2
dx
f(x) = sen x e�x2 |f(x)|  e�x2
x 2 RZ 1
1
e�x
2
dxZ 1
1
sen x e�x
2
dx ⇤
f(x) = e�x2 g(x) = �e�x2 h(x) = senx e�x2
� > 0
Z 1
1
1
x� + sen x
dx
1
x�+senx � 0 1x� > 0 x � 1
lim
x!1
x�
x� + sen x
= 1.
Z 1
1
1
x�
dx
� > 1Z 1
1
1
x� + sen x
dx � > 1 ⇤
� > 0
Z 1
0
1
x� + sen x
dx
0  1x�+senx  1x� x 2 [0, 1]
Z 1
0
1
x�
dx � < 1
Z 1
0
1
x� + sen x
dx
� < 1
� � 1 1x�+senx � 12x x 2 [0, 1]Z 1
0
1
x� + sen x
dx
� � 1 ⇤
� > 0
Z 3⇡
4
0
1
x�(1 + cos x)
dx
1
x�(1+cosx) � 0 1x� > 0 x 2 (0, 3⇡4 )
lim
x!0
x�
x�(1 + cos x)
=
1
2
.
Z 3⇡
4
0
1
x�
dx
� < 1Z 3⇡
4
0
1
x�(1 + cos x)
dx � < 1
⇤
Z 1
0
x2
3
p
x8 � 3x2 + 1 dx
Z 1
0
x2
2
p
x8 + 3x2 + 1
dxZ 2
0
x
(x� 2)3 dx
Z ⇡
2
0
1
x sen x
dxZ 1
0
e�xp
x
dx
Z 1
0
1
2
p
x+ 3
p
x4
dxZ 1
2
x+ 1p
x4 � x dx
Z 1
1
x
sen x+ xex
dx
F (x, y(x), y0(x), y00(x), ..., y(n)(x)) = 0, x 2 I,
x I y = y(x)
x y
y(k) k y = y(x)
y0(x) = f(x)
f
x2y00 � 2xy0 + y = ex
dy
dt
= y � t
(y � x)dy + 6xdx = 0
(y � x)dy
dx
+ 6x = 0
dy
dx
ut = uxx uxx + uyy = 0
@2v
@x2
=
@2v
@t2
uy = �vx
xy00 � (y0)3 + y = 0
y000 � 2xy0 + sen y = 0
(y � x)dy + 6xdx = 0
an(x)y(n) + an�1(x)y(n�1) + ...+ a1(x)y0 + a0(x)y = g(x).
y
ai x
y
x2y00�xy0+y = 0 x(4)�x000+4x00+x0+x = t+1 (cosx)y00�xy0+y = senx
xy00 � (y0)3 + y = 0 y = ty0 � ey0 (cos y)y0 + y = x2
n J ⇢ I
n J
y = cos x y00 + y = 0 R
y0 = � sen x y00 = � cosx
y00 + y = � cosx+ cosx = 0 8x 2 R
y = cos x R ⇤
• y = 0
y = 0
•
y0 = f(x) f
n n
•
(y0)2 + y2 = �1
I
y0 = ex
y(x) = ex + c x 2 R yp(x) = ex � 2 x 2 R
y = f(x) x 2 I
F (x, y) = 0
F (x, y) = x2 + y2 � 1 = 0 x 2 (�1, 1)
dy
dx
= �x
y
x
d
dx
(x2) +
d
dx
(y2) = 0 ) 2x+ 2y dy
dx
= 0 ) dy
dx
= �x
y
⇤
y =
p
1� x2 y = �p1� x2
x 2 (�1, 1)
n n y
x0
(
F (x, y(x), y0(x), y00(x), ..., y(n)(x)) = 0, x 2 I;
y(x0) = y0, y0(x0) = y1, y00(x0) = y2, ..., y(n�1)(x0) = yn�1.
m s0
v0
g
s(t)
t
F = ma = �mg
a(t) = s00(t)
s00(t) = �g
t s0(t) = �gt + c1
s(t) = �gt
2
2
+ c1t + c2
s0 = s(0) = �g ⇥ 0
2
2
+ c1 ⇥ 0 + c2 = c2 v0 = s0(0) = �g ⇥ 0 + c1 = c1
s(t) = s0 + v0t� gt
2
2
⇤
(
s00(t) = �g, t 2 R;
s(0) = s0, s0(0) = v0.
x2y0 + y = ex
(1 + y)y000 + xy = x
x00 + sen(t+ x) = x4
ety(4) � y00 + ty = 0
y00 + y = sec t, 0 < t < ⇡/2, y(t) = (cos t) ln(cost) + t sen t
y(4) + 4y000 + 3y = x, x 2 R, y(x) = e�x + x/3
r
y = tr t > 0
t2y00 � 4ty0 + 4y = 0
r
y = ert
y000 � 3y00 + 2y00 = 0
dy
dx
= f(x, y),
f : U ⇢ R2 ! R
f
f
f
x
dy
dx
= f(x).
f F I
y(x) = F (x)
I
f
dy
dx
= g(x)h(y).
g G
I 1/h H
J
H(y) = G(x) + C,
I y
x
H 0(y(x))y0(x) = G0(x).
H 1/h
1
h(y(x))
y0(x) = g(x).
y0(x) = h(y(x))g(x).
⌅
h dx
1
h(y)
dy = g(x)dx.
y x
h(y) 6= 0
h y0
y ⌘ y0
h(y) = 0
xy0 + y = 0.
dy
y
= �dx
x
.
x 6= 0 y 6= 0
ln |y| = � ln |x|+ C1,
C1 2 R
|y| = C|x| ,
C > 0 C 2 R/{0}
y = C/x (0,+1) (�1, 0)
C 2 R/{0}
y ⌘ 0 R ⇤
y0 = �x
y
.
y 6= 0
y dy = �x dx.
y2
2
= �x
2
2
+ C1,
C1 2 R
y2 + x2 = C,
y 6= 0
C > 0 ⇤
y0 = y2 � 4.
dy
y2 � 4 = dx.
y 6= �2 y 6= 2
1
4
{ln |y � 2|� ln |y + 2|} = x+ C1,
C1 2 R 4
|y � 2|
|y + 2| = Ce
4x,
C > 0 C 2 R/{0}
y
y =
2 + 2Ce4x
1� Ce4x ,
(� lnC4 ,+1) (�1,� lnC4 )
C > 0 C < 0 R
y ⌘ �2 y ⌘ 2 R
y ⌘ 2 C = 0
y ⌘ �2 ⇤
y0 =
p
y.
y � 0
dyp
y
= dx.
y 6= 0
2
p
y = x+ C,
C 2 R 2
y =
1
4
(x+ C)2.
x+C � 0
y (�C,+1)
y ⌘ 0 R
⇤
(1 + x)y dx+ (1� y)x dy = 0
y0 = 1+x
2
1+y2
sec2 ✓ tg � d✓ + sec2 � tg ✓ d� = 0
(x+ y2x) dx+ (y � x2y)x dy = 0
(x ln y) dydx = y
xy dydx = (1 + x
2) cossec y
y0 + y2 sen x = 0
xy0 =
p
1� y2
dy
dx =
x�e�x
y+ey
f : I1 ⇥ I2 ⇢ R2 ! R
I1 I2 (x0, y0) 2 I1 ⇥ I2
(
y0 = f(x, y),
y(x0) = y0.
J ⇢ I1 I1
u : J ⇢ R ! I2 ⇢ R u(x0) = y0
u0(x) = f(x, u(x))
x 2 J
(x0, y0) 2 I1 ⇥ I2 a
b R = [x0� a, x0 + a]⇥ [y0� b, y0 + b]
I1 ⇥ I2
f : I1 ⇥ I2 ! R M = max
(x,y)2R
f(x, y)
↵ = min{a, bM }
u : [x0 � ↵, x0 + ↵] ⇢ R! [y0 � b, y0 + b] ⇢ R
@f
@y
2 I1 ⇥ I2
u
y0 = �x
y
.
8<:y0 = �xy ,y(2) = 5 .
8<:y0 = �xy ,y(4) = �1 .
y0 = f(x, y) ,
f(x, y) = �xy f
D = {(x, y) 2 R2 : y 6= 0} f @f@y = xy2
D
(x0, y0) 2 D
C > 0
x2 + y2 = C,
x2 + y2 = C C = 1 C = 4 C = 9 C = 16 C = 25
(x0, y0) = (2, 5)
C =
p
29 y0 = 5 > 0
y(x) =
p
29� x2 ,
(�p29,p29)
(x0, y0) = (4,�1)
C =
p
17 y0 = �1 < 0
y(x) = �p17� x2 ,
(�p17,p17)
⇤
y0 =
p
y.
8<:y0 =
p
y ,
y(1) = 4 .
8<:y0 =
p
y ,
y(�1) = 0 .
y0 = f(x, y) , f(x, y) =
p
y
D = {(x, y) 2 R2 : y � 0}
f D @f@y =
1
2
p
yeD = D�{(x, y) 2 R2 : y = 0}
{(x, y) 2 R2 : y = 0}
(x0, y0) 2 D
(x0, y0) 2 eD (x0, 0)
C 2 R
y =
1
4
(x+ C)2 ,
(�C,+1)
y ⌘ 0 R
y = 14 (x+ C)
2 C = �3,�2, · · · , 4, 5
(x0, y0) = (1, 4) C = 3
y ⌘ 0
y(x) =
8<:14(x+ 3)2 x > �3 ,0 x  �3 ,
R
(x0, y0) = (�1, 0) C = 1
y ⌘ 0
y(x) =
8<:14(x+ 1)2 x > �1 ,0 x  �1 ,
y ⌘ 0
y(x) =
8<:14(x+ C)2 x > �C ,0 x  �C ,
C  1
⇤
(x, y) 2 R2
y0 + xy = 3 y(0) = 0
xy0 + y = 3 y(0) = 1
y0 = y2/3 y(0) = 0
y0 = e
x
y y(0) = 1
(1 + y2) dx+ (1 + x2) dy = 0 y(1) = 1
y0 = x
5/6
(1�t)3/2 y(0) = 1; y(1/2) = 1
y0 = y(1+2x)x(2+3y) y(1) = 1
y0 = y � y2 y(0) = 1; y(0) = 2; y(2) = 0
y0 = f(x, y) = F
⇣y
x
⌘
,
x 6= 0 F : I ⇢ R �! R
u(x) =
y(x)
x
,
y(x) = xu(x) y0(x) = u(x) + xu0(x)
y y0
xu0 = F (u)� u.
u x
y u = y/x
y0 =
y2 � x2
2xy
x > 0
y0 =
y2 � x2
2xy
=
1
2

y
x
� x
y
�
=
1
2
24y
x
� 1y
x
35
F (t) =
1
2

t� 1
t
�
u = y/x
u+ xu0 =
1
2

u� 1
u
�
, xu0 = �1
2

u2 + 1
u
�
Z
2uu0
u2 + 1
dx = �
Z
1
x
dx,
Z
1
v
dv = � lnx+ c
v = u2 + 1
ln |v| = � lnx + c
|v| = k
x
u u = y/x
y2 + x2 = kx
k > 0 ⇤
x
dy
dx
= y + xey/x x 6= 0
x
dy
dx
= y/x + ey/x
F (t) = t + et u = y/x y0 = u + xu0
u+ xu0 = u+ eu , xu0 = eu , e�uu0 = 1
x
.
�e�u = ln |x|+c y
e�y/x = � ln |x| + c = ln k|x| c = ln k k > 0
y = �x ln
✓
ln
k
|x|
◆
⇤
y x F
f : D ⇢ R2 ! R f(x, y) = F
⇣y
x
⌘
f(tx, ty) = f(x, y)
8t 6= 0 (tx, ty) 2 D
f(x, y) = F
⇣y
x
⌘
f(tx, ty) = F
✓
ty
tx
◆
= F
⇣y
x
⌘
= f(x, y)
f(x, y) = f(x1, x
y
x
) = f(1,
y
x
) 8x 6= 0
F (t) = f(1, t) t 2 I I f(x, y) = F
⇣y
x
⌘
⌅
8<: y0 =
x+ y
x� y , x > 0
y(1) = 2
f(tx, ty) =
tx+ ty
tx� ty =
x+ y
x� y = f(x, y)
u = y/x
u+ xu0 =
1 + u
1� u ,
1� u
u2 + 1
u0 =
1
x
arctg u� 1
2
ln(u2 + 1) = ln x+ c x > 0
arctg y/x� 1
2
ln(y2 + x2) = c
y(1) = 2 c = arctg 2 � 1
2
ln 5
arctg y/x�1
2
ln(y2+x2) = arctg 2�1
2
ln 5
⇤
y0 =
ax+ by
cx+ dy
a, b, c, d c d
f(x, y) =
ax+ by
cx+ dy
= f(tx, ty)
dy = y0(x)dx e f(x, y) = �M(x, y)
N(x, y)
,
y0(x) = f(x, y) = �M(x, y)
N(x, y)
,
dy = �M(x, y)
N(x, y)
dx
M(x, y)dx+N(x, y)dy = 0.
dx
↵ 2 R M(tx, ty) = t↵M(x, y) N(tx, ty) =t↵N(x, y)
(xy + y2)dx� x2dy = 0 x 6= 0
M(tx, ty) = txty + (ty)2 = t2(xy + y2) = t2M(x, y)
N(tx, ty) = �(tx)2 = t2(�x2) = t2N(x, y),
u = y/x dy du dx
dy = udx+ xdu
[xux+ (ux)2]dx� x2(udx+ xdu) = 0, x2[u2dx� xdu] = 0
, u2dx� xdu = 0, dx
x
=
du
u2
.
x u
ln |x| = �1
u
+ c , u = 1
c� ln |x| y =
x
c� ln |x|
y = 0 y/x = u = 0
⇤
M(x, y)dx + N(x, y)dy = 0
v =
x
y
y x
v M(x, y)
N(x, y) u =
y
x
y2dx� (x2 + xy + y2)dy = 0
v = x/y dx = vdy + ydv
y2(vdy + ydv)� (y2v2 + vy2 + y2)dy = 0, ydv � (v2 + 1)dy = 0.
dv
v2 + 1
=
dy
y
arctg v = ln |y|+ c arctg(x/y) = ln |y|+ c.
⇤
u =
y
x
1 + u+ u2
u+ u3
du = �dx
x
u
(x+ y)y0 = y � x
xy2y0 = y3 � x3
ydx+ x(lnx� ln y � 1)dy = 0
xdy = (y +
p
x2 � y2)dx, x > 0
(x2 + 2y2)dx = xydy, y(�1) = 1
(x+
p
xy)
dy
dx
+ x� y = x�1/2y3/2, y(1) = 1
a(x)y0 + b(x)y = c(x),
a, b c c
(x0, y0) a, b
c a(x0) 6= 0
a(x) 6= 0 a(x)
y0 + P (x)y = Q(x),
P (x) = b(x)a(x) Q(x) =
c(x)
a(x)
a(x) ⌘ 0 y
b(x)y(x) = c(x)
a
yh yp
y = yh + yp.
y0 + P (x)y = 0.
yh(x) = c e
� R P (x)dx,
c 2 R
c(·)
yp(x) = c(x)e�
R
P (x)dx
c c(·)
yp(x) = c(x)e
� R P (x)dx y0p(x) = [c0(x)� P (x)c(x)]e� R P (x)dx.
yp
[c0(x)� P (x)c(x)]e�
R
P (x)dx + P (x)c(x)e�
R
P (x)dx = Q(x).
c0(x) = Q(x)e
R
P (x)dx.
c(x) =
Z
Q(x)e
R
P (x)dx dx+ C.
C = 0
yp(x) = e
� R P (x)dx Z Q(x)eR P (x)dx.
y = Ce�
R
P (x)dx + e�
R
P (x)dx
Z
e
R
P (x)dxQ(x)dx,
C
�(x) = e
R
P (x)dx,
(y(x)e
R
P (x)dx)0 = Q(x)e
R
P (x)dx.
y(x)e
R
P (x)dx =
Z
Q(x)e
R
P (x)dx dx+ C ,
C e
R
P (x)dx
y(x) = e�
R
P (x)dx
nZ
Q(x)e
R
P (x)dx dx+ C
o
,
�(x)
y0 � y = 2xe2x.
y0 � y = 0.
yh(x) = c e
x,
c 2 R
yp(x) = c(x)e
x y0p(x) = [c
0(x) + c(x)]ex.
yp y0p
[c0(x) + c(x)]ex � c(x)ex = 2xe2x c0(x) = 2xex.
yp(x) = 2xe
2x � 2e2x.
y = Cex + 2xe2x � 2e2x,
C ⇤
y0 � tg(x) y = sen(x).
x
cos(x) 6= 0
�(x) = e�
R
tg xdx = eln cosx = cos x ,
cos(x)y0 � sen(x)y = sen(x) cos(x) .
(y(x) cos(x))0 = sen(x) cos(x) .
y(x) cos(x) = �1
2
cos2(x) + C ,
C cos(x)
y(x) = C sec(x)� 1
2
cos(x).
⇤
x3y0 + 4x2y = e�x.
x 6= 0
y0 + 4x�1y = x�3e�x,
�(x) = e4
R
x�1dx = e4 ln |x| = x4 .
x4y0 + 4x3y = xe�x ,
(y(x)x4)0 = xe�x.
y(x)x4 = �e�x � xe�x + C ,
C x4
y(x) = Cx�4 � x�4[1 + x]e�x.
⇤
(1 + x2)y0 + 4xy = (1 + x2)�2 .
y0 +
4x
1 + x2
y = (1 + x2)�3 ,
�(x) = e
R 4x
1+x2
dx = e2 ln (1+x
2) = (1 + x2)2 .
(1 + x2)2y0 + 4x(1 + x2)y = (1 + x2)�1 ,
(y(x)(1 + x2)2)0 =
1
1 + x2
.
y(x)(1 + x2)2 = arctg(x) + C ,
C (1+x2)2
y(x) = C(1 + x2)�2 + (1 + x2)�2 arctg(x).
⇤
x(0) = 1
x0(t)� 3x(t) = 5
3x0(t) + 2x(t) + 16 = 0
x0(t) + 2x(t) = t2
tx0(t) + 2x(t) + t = 0 t 6= 0
x0(t)� x(t)t = t t > 0
x0(t)� tx(t)t2�1 = t t > 1
x0(t)� 2x(t)t + 2t2 = 0 t > 0
8<:x0(t)G(t) + x(t)G0(t) = tG0(t)x(t0) = t0 ,
G
G
z = f(x, y)
R xy
df dz
df = fxdx+ fydy.
M(x, y)dx+N(x, y)dy = 0
f(x, y) C2 R ⇢ R2
M(x, y) = fx(x, y) e N(x, y) = fy(x, y);
R
fx + fy
dy
dx
= 0,
d
dx
f(x, y(x)) = fx + fy
dy
dx
= 0,
f(x, y(x)) = c
f(x, y)
f(x, y) = c
f
f(x, y) = c
f
M,N : I1 ⇥ I2 ! R C1 I1, I2
@
@y
M(x, y) =
@
@x
N(x, y)
8(x, y) 2 I1 ⇥ I2
•
f C2
•
f(x, y)
f
x
y
4x3ydx+ (x4 + 2y)dy = 0,
R2
M(x, y) = 4x3y N(x, y) = x4 + 2y
@
@y
M(x, y) = 4x3 e
@
@x
N(x, y) = 4x3.
R2
f df = 0
x
f(x, y) =
Z
4x3y dx+ g(y) = 4y
Z
x3 dx+ g(y) = x4y + g(y);
g y
f g x
fy(x, y) = x
4 + g0(y) = x4 + 2y,
g0(y) = 2y
g(y) = y2 + k,
k
f(x, y) = x4y + y2 + k.
f(x, y) = x4y+y2+k = c
c k
x4y + y2 = c c 2 R
⇤
(e3y � y cosxy)dx+ (3xe3y � x cosxy + y3)dy = 0,
R2
My = 3e3y�cosxy+xy sen xy Nx = 3e3y�cosxy+xy sen xy
x
f(x, y) =
Z
e3y � y cosxy dx+ g(y) = xe3y � sen xy + g(y).
fy(x, y) = 3xe
3y � x cosxy + g0(y) = 3xe3y � x cosxy + y3,
g0(y) = y3
g(y) =
y4
4
+ k.
f(x, y) = xe3y � sen xy + y
4
4
.
xe3y � sen xy + y
4
4
= c.
⇤
8<: y0 =
(x+ y)2
1� x2 � 2xy ;
y(1) = 1.
dy
dx
=
(x+ y)2
1� x2 � 2xy ) (x+ y)
2dx+ (x2 + 2xy � 1)dy = 0.
My = 2(x + y) = 2x + 2y Nx = 2x + 2y
x
f(x, y) =
(x+ y)3
3
+ g(y).
fy(x, y) = (x+ y)
2 + g0(y) = x2 + 2xy � 1,
g0(y) = �1� y2
g(y) = �y � y
3
3
+ k.
f(x, y) =
(x+ y)3
3
� y � y
3
3
.
(x+ y)3
3
� y � y
3
3
= c.
(1 + 1)3
3
� 1� 1
3
3
= c) c = 8
3
� 1� 1
3
=
4
3
.
(x+ y)3
3
� y � y
3
3
=
4
3
.
⇤
(x+ y)dx+ (x lnx)dy = 0, x > 0, y 2 R.
My = 1 6= Nx = ln x+ 1.
I(x) =
1
x
(1 +
y
x
)dx+ lnxdy = 0, x > 0, y 2 R,
I
M(x, y)dx+N(x, y)dy = 0;
I(x, y)
I
(IM)y = (IN)x,
IyM + IMy = IxN + INx,
I
x
IMy = I
0N + INx,
My �Nx
N
x I
I 0
I
=
My �Nx
N
.
x
I = I(x)
I(x)
I(x) = e
R My�Nx
N dx.
x
I
y
Nx �My
M
y
I(y) = e
R Nx�My
M dy.
(xy + x2 + 1)dx+ x2dy = 0, x > 0.
My �Nx
N
= �1
x
x
x
I(x) = e
R My�Nx
N dx = e�
R dx
x = e� lnx =
1
x
.
(y + x+
1
x
)dx+ xdy = 0, x > 0.
f(x, y) df = 0
x
f(x, y) = yx+
x2
2
+ lnx+ g(y).
fy(x, y) = x+ g
0(y) = x.
g0(y) = 0 g
yx+
x2
2
+ lnx = c.
y =
1
x
[c� x
2
2
� lnx], x > 0.
⇤
(cosx+ 1)dx+ (
x+ sen x
y
� 2)dy = 0, y > 0.
Nx �My
M
=
1
y
y
y
I(y) = e
R Nx�My
M dy = y.
(y cosx+ y)dx+ (x+ sen x� 2y)dy = 0.
f(x, y) df = 0
x
f(x, y) = yx+ y sen x+ g(y).
y
fy(x, y) = x+ sen x+ g
0(y) = x+ sen x� 2y.
g0(y) = �2y g(y) = �y2 + k
yx+ y sen x� y2 = c,
y > 0 ⇤
•
y = 0 y = 0
•
I(x)
y0 + p(x)y = q(x), [p(x)y � q(x)]dx+ dy,
M(x, y) = p(x)y � q(x) N(x, y) = 1
• I(x, y)
I(x, y)
(2x� 1)dx+ (7y + 4)dy = 0
(4x3 + 4xy)dx+ (2x2 + 2y � 1)dy = 0
(4x� 8y3)dy + (5x+ 4y)dx = 0
(y2x+ x)dx� (yx2)dy = 0
cosx cos yy0 + (tg x� sen x sen y) = 0�
1 + lnx+ yx
�
dx = (1� lnx)dy x > 0
dy
dx
=
y
x
(ln y � lnx) x > 0
(ex � y)dx+ (2� x+ yey)dy = 0 y(0) = 1
(x+ y)dx+ (x lnx)dy = 0, x > 0 y(e) = 1
(x2 + 2xy � y2)dx+ (y2 + 2xy � x2)dy = 0
I(x, y) = (x+ y)�2
M(x, y)
M(x, y)dx+
✓
xexy + 2xy +
1
x
◆
dy = 0;
h(y)dy�g(x)dx = 0
h g
↵ 2 R ↵ /2 {0, 1}
y0 + P (x)y = Q(x)y↵,
Q(x) = 0 ↵ 2 {0, 1}
y ⌘ 0
↵ > 0 y↵
y0
y↵
+
P (x)
y↵�1
= Q(x).
w = y1�↵ w0(x) = (1 � ↵)y�↵y0
w
w0 + (↵� 1)P (x)w = (↵� 1)Q(x),
w = y1�↵
xy0 + y = x3y3 x 6= 0.
y0 +
1
x
y = x2y3.
w = y�2 w0 = �2y�3y0
w0 � 2
x
w = �2x2.
x�2
[x�2w]0 = �2.
x�2w = �2x+ C,
C 2 R w = y�2
y2 =
1
x2(C � 2x) .
⇤
2xyy0 � y2 + x = 0 x 6= 0.
y0 � 1
x
y = �1
2
y�1.
w = y2 w0 = 2yy0
w0 � 1
x
w = �1.
x�1
[x�1w]0 = �1
x
.
x�1w = � ln |x|+ C,
C 2 R w = y2
y2 = x(C � ln |x|).
⇤
y0 = P (x)y2 +Q(x)y +R(x)
P (x) = 0
R(x) = 0
1a)
yp
yp z = y � yp
z0 = y0� y0p) z yp
z0 + y0p = P (x){z2 + 2zyp + y2p}+Q(x){z + yp}+R(x).
yp
y0p = P (x)y
2
p +Q(x)yp +R(x),
z0 = P (x)z2 + [Q(x) + 2P (x)yp]z.
z
y = z + yp
y = yp + 1/w
2a)
v(x) = P (x)y(x) v0(x) = P (x)y0(x) + P 0(x)y(x)
v
v0 = v2 +
h
Q(x) +
P 0(x)
P (x)
i
v + P (x)R(x).
v = �u0u v0 = (u
0)2�u00u
u2 u
�u
00
u
= �
h
Q(x) +
P 0(x)
P (x)
iu0
u
+ P (x)R(x).
u
u00 �
h
Q(x) +
P 0(x)
P (x)
i
u0 + P (x)R(x)u = 0.
yp = x
y0 =
1
x
y2 � 2(1� 1
x
)y + x� 1
yp = x
1 =
1
x
x2 � 2(1� 1
x
)x+ x� 1.
y = z+x
y0 = z0 + 1
z0 � 2
x
z =
z2
x
,
w = z�1 w0 = �z�2z0
w0 +
2x
w = �2
x
.
x2
[x2w]0 = �2x.
x2w = �x2 + C,
C 2 R w = z�1 y = z + x
y =
x2
C � x2 + x.
⇤
yp = ex
y0 � (1 + 2ex)y + y2 = �e2x
yp = ex
ex � (1 + 2ex)ex + e2x = �e2x.
y = z + ex
y0 = z0 + ex
z0 � z = �z2.
w = z�1 w0 = �z�2z0
w0 + w = 1.
ex
[exw]0 = ex.
exw = ex + C,
C 2 R w = z�1 y = z + ex
y =
1
1 + Ce�x
+ ex.
⇤
y = xy0 + f(y0),
f
x
y0 = y0 + xy00 + f 0(y0)y00,
y00(x+ f 0(y0)) = 0.
y00 = 0 x+ f 0(y0) = 0.
y0 = C x = �f 0(y0) y0 = C
y = xC + f(C),
C x = �f 0(y0)
y0 = p
(
x = �f 0(p),
y = xp+ f(p),
)
(
x = �f 0(p),
y = �f 0(p)p+ f(p).
y = xy0 � (y0)2
x
y00(x� 2y0) = 0.
y00 = 0 x� 2y0 = 0.
y = Cx� C2,
C 2 R (
x = 2p,
y = 2p2 � p2 = p2.
p =
x
2
y =
⇣x
2
⌘2
=
x2
4
y =
x2
4
y = Cx� C2
⇤
y � xy0 = � ln y0
y = xy0 � ln y0
x
y = Cx� lnC,
C > 0
8<: x =
1
p
,
y = 1� ln p.
y = 1 + ln x
y = 1 � ln p = 1 � ln
✓
1
x
◆
= 1 � ln 1 + lnx = 1 + lnx x > 0
y = 1 + lnx y = Cx� lnC C > 0
⇤
y0 + xy = x3y3
y0(x2y3 + xy) = 1
(2� y lnx)dx+ xdy = 0
y � y0 cosx = y2 cosx(1� sen x)
y = xy0 + 1� ln y0
y(y0)2 = x(y0)3 + 1
yp
y0 = y2 + 2y � 15 yp(x) = �3
y0 = y2 � yx � 25x2 yp(x) = 5x
y0 = cossec2 x+ y cotg x+ y2 yp(x) = � cotg x
y0 = y2 + 8xy + 16x2 � 4 yp(x) = �4x
F
y0 = f(x, y).
(x0, y0)
f(x0, y0)
y0(x0)
(x0, y0) (x0, y0)
F
� 1
f(x0, y0)
f(x0, y0) 6= 0
F
y0 = � 1
f(x0, y0)
.
F
G(x, y, c) = 0,
c
x
x
Gx +Gyy
0 = 0) y0 = �Gx
Gy
.
c x y
y0 = �Gx(x, y(x), c(x, y))
Gy(x, y(x), c(x, y))
= f(x, y),
c x y
y = cx2.
x
y0 = 2cx.
c =
y
x2
y0 =
2y
x
,
⇤
x2 + y2 = c,
c > 0
2x+ 2yy0 = 0,
y0 = �x
y
.
y0 =
y
x
.
y = kx ⇤
y =
c
x
,
(1, 2)
x
y0 = � c
x2
,
c = yx
y0 = �y
x
.
y0 =
x
y
.
y2 � x2 = c,
(1, 2)
c = 3 y2�x2 = 3
⇤
(
y0(t) = ky(t),
y(t0) = y0,
k
y(t) = y0e
k(t�t0).
y(t)
k > 0 y(t) k < 0
dy
dt
y(t)
y(t) t
t0 = 0
y(t) = y0ekt
k
2y0 = y(1) = y0e
k,
ek = 2 k = ln 2
y(t) = y0e
t ln 2 = y02
t.
t
3y0 = y(t) = y0e
t ln 2,
et ln 2 = 3 t =
ln 3
ln 2
' 1, 58h
⇤
k
y(t) y(0) = y0
y(t) = y0ekt
k
y0
2
= y0e
5700k,
y0
k = � ln 2
5700
.
0, 02y0 t
0, 02y0 = y0e
� ln 25700 t,
y0
t = �5700ln 0, 02
ln 2
' 32.170anos.
⇤
T (t)
Tm
dT
dt
= k(T � Tm),
k
y(t) = T (t) � Tm Tm
dy
dt
=
dT
dt
T (t) = Tm+ cekt t = 0
T0
T (t) = Tm + (T0 � Tm)ekt,
T (0) = 180 Tm = 20
T (t) = 20 + (180 � 20)ekt = 20 + 160ekt t
120 = T (4) = 20 + 160e4k k =
1
4
ln
5
8
25 = 20 + 160e
t
4 ln
5
8 t = 4 ln
5
160
/ ln
5
8
= �20 ln 2/ ln 5
8
' 29, 5
⇤
m
v(0) = v0
v(t)
m
dv
dt
= mg � kv2,
k > 0 kv2
dv
dt
v(t) t!1
m
dv
dt
= (
p
mg �
p
kv)(
p
mg +
p
kv).
Z
mdv
(
p
mg �pkv)(pmg +pkv) =
Z
dt,
r
mg
k
ln
�����
p
mg + v
p
k
p
mg � vpk
����� = 2t+ C.
p
mg + v
p
k
p
mg � vpk = Ce
2t
q
k
mg .
v t
v(t) =
r
mg
k
24Ce2tq kmg � 1
Ce
2t
q
k
mg + 1
35 .
v(0) = v0
v(t) =
r
mg
k
264(
p
mg+
p
kv0p
mg�pkv0 )e
2t
q
k
mg � 1
(
p
mg+
p
kv0p
mg�pkv0 )e
2t
q
k
mg + 1
375 .
t ! 1 v(t) !
r
mg
k
⇤
Q0
y = cxex y = c sen x
c > 0 y = f(x)
y = c lnx f f(1) = 2
x+ y = cey
y2 = c(2x+ c)
n � 2
n � 2
n � 2
an(x)y
(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g(x),
y(k) k y x a0, · · · , an, g
g
1
yh yp
y = yh + yp.
a0, · · · , an, g I
an 8<:an(x)y(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g(x),y(x0) = y0, y0(x0) = y1, · · · , y(n�1)(x0) = yn�1,
x0 2 I I
n � 2
an(x)y
(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = 0.
y1, y2, · · · , yk I
c1y1 + c2y2 + · · ·+ ckyk,
I
N � 2
f1(x) = 1 + sen x f2(x) = 1 + cosx
y00 + y = 1 f1 + f2 2f1
y1, y2, · · · , yk I
c1y1(x) + c2y2(x) + · · ·+ ckyk(x) = 0,
x 2 I c1 = c2 = · · · = 0
Rn
n
y1, y2, · · · , yk k � 1
I I W [y1, · · · , yk](x0) 6= 0
x0 2 I
W [y1, · · · , yk](x) =
����������
y1(x) y2(x) · · · yk(x)
y
0
1(x) y
0
2(x) · · · y0k(x)
· · ·
y(k�1)1 (x) y
(k�1)
2 (x) · · · y(k�1)k (x)
����������
,
y1, y2, · · · , yk x
I
n y1, y2, · · · , yn I
W [y1, · · · , yn](x0) 6= 0 x0 2 I
y(x) = c1y1(x) + c2y2(x) + · · ·+ cnyn(x),
I y1, y2, · · · , yn
N � 2
R
{2x� 3, x2 + 1, 2x2 � x} {2x� 3, 2x2 + 1, 3x2 + x}.
W (2x� 3, x2 + 1, 2x2 � x) =
�������
2x� 3 x2 + 1 2x2 � x
2 2x 4x� 1
0 2 4
�������
= (2x� 3)
�����2x 4x� 12 4
������ 2
�����x2 + 1 2x2 � x2 4
�����
= 2(2x� 3)� 2(4 + 2x) = �10 6= 0.
W (2x+ 3, 2x2 + 1, 3x2 � x) =
�������
2x� 3 2x2 + 1 3x2 � x
2 4x 6x� 1
0 4 6
�������
= (2x+ 3)
�����4x 6x� 14 6
������ 2
�����2x2 + 1 3x2 � x4 6
�����
= 4(2x+ 3)� 2(6 + 4x) = 0.
[2x+ 3]� 3[2x2 + 1] + 2[3x2 � x] = 0.
⇤
W (x3, |x3|)
{x3, |x3|}
|x3| |x3|0
|x3| =
8<:x3 x > 0�x3 x  0 , |x3|0 =
8<:3x2 x > 0�3x2 x  0.
W (x3, |x3|) =
����� x3 x33x2 3x2
����� = 0, x > 0,
W (x3, |x3|) =
����� x3 �x33x2 �3x2
����� = 0, x  0.
W (x3, |x3|) = 0 x 2 R
c1 c2
c1x
3 + c2|x3| = 0,
x = 1 x = �1
c1 + c2 = 0,
�c1 + c2 = 0,
c1 = c2 = 0 {x3, |x3|}
⇤
N � 2
xy000 � y00 = 0 {y1 = 1, y2 = x, y3 = x3}
x3y000 + x2y00 � 2xy0 + 2y = 0 {y1 = x, y2 = x2, y3 = 1/x}
{y1 = 1, y01 = 0, y001 = 0, y0001 = 0} x[0]� [0] = 0
{y2 = x, y02 = 1, y002 = 0, y0002 = 0} x[0]� [0] = 0
{y3 = x3, y03 = 3x2, y003 = 6x, y0003 = 6} x[6]� [6x] = 0
{y1 = x, y01 = 1, y001 = 0, y0001 = 0} x3[0] + x2[0]� 2x[1] + 2[x] = 0
{y2 = x2, y02 = 2x, y002 = 2, y0002 = 0} x3[0]+x2[2]�2x[2x]+2[x2] = 0
{y3 = x�1, y03 = �x�2, y003 = 2x�3, y0003 = �6x�4}
x3[�6x�4] + x2[2x�3]� 2x[�x�2] + 2[x�1] = (�6 + 2 + 2 + 2)x�1 = 0
W (1, x, x3) =
�������
1 x+ 1 x3
0 1 3x2
0 0 6x
������� =
�����1 3x20 6x
����� = 6x.
W (x, x2, x�1) =
�������
x x2 x�1
1 2x �x�2
0 2 2x�3
������� = x
�����2x �x�22 2x�3
������
�����x2 x�12 2x�3
����� = 6x�2.
(�1, 0) (0,1) ⇤
8<:xy000 � y00 = 0,y(x0) = y0, y0(x0) = y1, y00(x0) = y2.
8<:x3y000 + x2y00 � 2xy0 + 2y = 0,y(x0) = y0, y0(x0) = y1, y00(x0) = y2.
x0 6= 0
y(x) = c1 + c2x+ c3x
3
(�1, 0) (0,1)
c1 =
y0
6x0
, c2 = y1 � y2
2x0
c3 = y0 � x0y1 + 1
2
y2 +
1
6
y2x
2
0.
x0 = 0
{x0 = 0, y0 = 1, y1 = 1, y2 = 0},
y(x) = 1 + x R
{x0 = 0, y0 = 0, y1 = 0, y2 = 1},
x0 6= 0
y(x) = c1x+ c2x
2 +
c3
x
(�1, 0) (0,1)
c1 =
y0
x0
� 3
4
x0y2, c2 =
y2
2
+
y1
3x0
� y0
3x20
c3 =
1
3
x0y0� 1
3
x20y1+
1
6
x30y2.
N � 2
x0 = 0
{x0 = 0, y0 = 0, y1 = 1, y2 = 1},
y(x) = x + x2 R
{x0 = 0, y0 = 1, y1 = 1, y2 = 1}
⇤
xy000 + (senx)y00 + 3y = cos x
x(x� 1)y(4) + exy00 + 4x2y = 0
(x� 1)y(4) + (x+ 1)y00 + (tg x)y = 0
(x2 � 4)y(4) + x2y000 + 9y = 0
y00 � y0 � 12y = 0 {y1 = e�3x, y2 = e4x} R
y00 � 2y0 + 5y = 0 {y1 = ex cos 2x, y2 = ex sen 2x} R
x2y00 + xy0 + y = 0 {y1 = cos(lnx), y2 = sen(lnx)} (0,1)
x3y000+6x2y00+4xy0�4y = 0 {y1 = x, y2 = x�2, y3 = x�2 lnx} (0,1)
x2y00 � 4xy0 + 6y = 0.
{y1 = x3, y2 = |x3|} {z1 = x2, z2 = x3}
R
y(x) = c1x
3 + c2|x3| y(x) = c1x2 + c2x3,
R
y00 + P (x)y0 +Q(x)y = 0,
P Q I
y1 I y1(x) 6= 0 8x 2 I
y2 y1 y2 = �(x)y1(x)
� C2 I {y1, y2}
� y2 = �(x)
y02 = �
0y1 + �y01 e y
00
2 = �
00y1 + 2�0y01 + �y
0
1 + �y
00
1 ,
�(x)
y002 + P (x)y
0
2 +Q(x)y2 = �(x)(y
00
1 + P (x)y
0
1 +Q(x)y1)+
+�00y1 + �0(x)(2y01(x) + P (x)y1(x)) = 0.
y1
y1�
00 + (2y01 + Py1)�
0 = 0.
y1 �0u u
y1u
0 + (2y01 + Py1)u = 0.
u(x) = c1e
� R 2y01+Py1y1 dx = c1e�
R
P (x)dx
y21
.
u = �0
�(x) =
Z
c1e�
R
P (x)dx
y21
dx+ c2.
c1 = 1 c2 = 0
�(x)
�(x) =
Z
e�
R
P (x)dx
y21
dx.
y2(x) = y1(x)
Z
e�
R
P (x)dx
y21
dx.
{y1, y2}
xy00�2y0+4(1�x)y = 0 y1(x) = e2x
y(1) = 1, y0(1) = 0
�
y2(x) = �(x)e
2x
x�00(x) + (4x� 2)�0(x) = 0,
�00(x) + (4� 2
x
)�0(x) = 0,
u(x) = �0(x)
u0(x) + (4� 2
x
)u(x) = 0.
e
R
4� 2xdx =
e4x
x2
u(x) = c1x2e�4x c1 = 1
�0 = u(x) �(x) = �x
2e�4x
4
� e
�4x
8
y(x) = k1e2x + k2(2x2e�2x + e�2x)
y(1) = 1, y0(1) = 0 k1 k2
y(x) =
e2x�2
4
+
e�2x+2
4
(2x2 + 1).
⇤
y1(x) = x4 x2y00�7xy0+16y = 0 x > 0
y00 � 7
x
y0 +
16
x2
y = 0
y2(x) = x
4
Z
e
R 7
xdx
x8
dx = x4
Z
e7 lnx
x8
dx = x4 lnx.
y(x) = c1x4 + c2x4 lnx x > 0 ⇤
y00 + y0 = 0
y0 = u
y1 = 1 y2 = �e�x
y(x) = c1 + c2e�x ⇤
y1
y00 � 2y0 = 0, y1(x) = 1
y00 � y = 0, y1(x) = ex
y00 � 4y0 + 4y = 0, y1(x) = e2x
y00 + 6y0 + 9y = 0, y1(x) = e�3x
x2y00 � 6y = 0, y1(x) = x3 x > 0
x2y00 � 3xy0 + 5y = 0, y1(x) = x2 cos(lnx)
y00 � 4y = 2
y1(t) = e�2t
y00 + by0 + cy = 0 b2 � 4c = 0
y1(t) = e�bt/2
y1
n
n
n
any
(n) + an�1y(n�1) + · · ·+ a1y0 + a0y = 0,
a0, · · · , an
n = 2
a2y
00 + a1y0 + a0y = 0.
y = e↵x ↵ 2 R
y e↵x
[a2↵
2 + a1↵+ a0]e
↵x = 0.
e↵x
a2↵
2 + a1↵+ a0 = 0.
P (z) = a2z
2 + a1z + a0.
� = a21 � 4a0a2 � > 0,� = 0,� < 0
� > 0
↵1 =
�a1+
p
�
2a2
↵2 =
�a1�
p
�
2a2
y1 = e
↵1x y2 = e
↵2x,
W [y1, y2](x) =
����� e↵1x e↵2x↵1e↵1x ↵2e↵2x
����� = (↵2 � ↵1)e(↵1+↵2)x 6= 0,
x 2 R
y(x) = c1e
↵1x + c2e
↵2x
N
� < 0
�1 = a+ bi �2 = a� bi a = �a12a2 b =
p��
2a2
w1 = e
�1x w2 = e
�2x,
w1 w2
ei✓ = cos(✓) + i sen(✓)
w1 = e
ax[cos(bx) + i sen(bx)] w2 = e
ax[cos(bx)� i sen(bx)].
1
2
w1 +
1
2
w2 = e
ax cos(bx)
1
2i
w1 � 1
2i
w2 = e
ax sen(bx).
y1 = e
ax cos(bx) y2 = e
ax sen(bx),
����� eax cos(bx) eax sen(bx)aeax cos(bx)� beax sen(bx) aeax sen(bx) + beax cos(bx)
����� = beax 6= 0,
x 2 R
y(x) = c1e
ax cos(bx) + c2e
ax sen(bx)
� = 0 2
↵ = � a12a2
y1 = e
↵x,
y2
y1
c(·)
y2(x) = c(x)e
↵x
y02 y
00
2
y02(x) = [c
0(x) + ↵c(x)]e↵x ; y002(x) = [c
00(x) + 2↵c0(x) + c(x)↵2]e↵x.
y2
a2[c
00(x) + 2↵c0(x) + c(x)↵2] + a1[c0(x) + ↵c(x)] + a0c(x) = 0,
a2[c
00(x) + 2↵c0(x)] + a1c0(x) = 0.
c0 c00 c
d(x) = c0(x)
d0(x) + [2↵+ a1/a2]d(x) = 0.
↵ = � a12a2 d0(x) = 0 d
c
c(x) = mx+ k,
m k m = 1 d = 0
y2(x) = xe
↵x
����� e↵x xe↵x↵e↵x e↵x(1 + ↵x)
����� = e↵x 6= 0,
x 2 R y1
N
y(x) = c1e
↵x + c2xe
↵x
n � 3
n � 3
n = 2 y e↵x
y = e↵x ↵
P (x) = anz
n + an�1zn�1 + · · ·+ a1z + a0.
n P
P
↵ P k
{e↵x, xe↵x, · · · , xk�1e↵x}
�1 = a+ bi �2 = a� bi
P k
{sen(ax)ebx, cos(ax)ebx, · · · , xk�1 sen(ax)ebx, xk�1 cos(ax)ebx}
�1 = �1 �2 = 5
�1 =
p
3� 2i �2 =
p
3 + 2i
�1 = 1/2 �2 = �2 �3 = 3
�1 = �1 �2 = �2 + i �3 = �2� i
�1 = �2 = 2 �3 = 3
�1 = �2 = �3 = 1/4
�1 = �2 �2 = �1 �3 = 5 �4 = 7
�1 = �2 �2 = �1 �3 = 5 + 4i �4 = 5� 4i
�1 = �2 = �2 + 3i �3 = �4 = �2� 3i
�1 = �2 = �3 = �4 = 1/3,�5 = 3
y(x) = c1e�x + c2e5x
y(x) = c1e
p
3x cos(2x) + c2e
p
3x sen(2x)
y(x) = c1ex/2 + c2e�2x + c3e3x
y(x) = c1e�x + c2e�2x cos(x) + c3e�2x sen(x)
y(x) = c1e2x + c2xe2x + c3e3x
N
y(x) = c1ex/4 + c2xex/4 + c3x2ex/4
y(x) = c1e�2x + c1e�x + c1e5x + c1e7x
y(x) = c1e�2x + c1e�x + c3e5x cos(4x) + c4e5x sen(4x)
y(x) = c1e�2x cos(3x)+c2xe�2x cos(3x)+c3e�2x sen(3x)+c4xe�2x sen(3x)
y(x) = c1ex/3 + c2xex/3 + c3x2ex/3 + c4x3ex/3 + c5e3x
⇤
8<:y(3) + y0 = 0;y(0) = 0, y0(0) = 0, y00(0) = 2.
8<:4y(3) + y0 + 5y = 0;y(0) = 2, y0(0) = 1, y00(0) = �1.
8<:y(4) � 4y(3) + 4y00 = 0;y(1) = �1, y0(1) = 2, y00(1) = 0, y(3)(1) = 0.
z3 + z = z(z2 + 1),
�1 = 0 �1 = i �1 = �i
y(x) = c1 + c2 cosx+ c3 sen x ,
y0(x) = �c2 sen x+ c3 cosx ,
y00(x) = �c2 cosx� c3 sen x .
2641 1 00 0 1
0 �1 0
375
0B@c1c2
c3
1CA =
0B@00
2
1CA .
c1 = 0 c2 = 0 c3 = 2
y(x) = 2 senx , x 2 R .
4z3 + z + 5 = (z + 1)(4z2 � 4z + 5)
�1 = �1 �1 = 1/2 + i �1 = 1/2� i
y(x) = c1e
�x + c2ex/2 cosx+ c3ex/2 sen x ,
y0(x) = �c1e�x + c2ex/2[1/2 cos x� sen x] + c3ex/2[1/2 senx+ cosx] ,
y00(x) = c1e�x + c2ex/2[�3/4 cos x� sen x] + c3ex/2[�3/4 senx+ cosx] .
264 1 1 0�1 1/2 1
1 �3/4 1
375
0B@c1c2
c3
1CA =
0B@ 21
�1
1CA .
c1 = 2/13 c2 = 24/13
c3 = 3/13
y(x) = 2/13e�x + 24/13ex/2 cosx+ 3/13ex/2 sen x .
z4 � 4z3 + 4z2 = z2(z2 � 4z + 4) = z2(z � 2)2,
N
�1 = �2 = 0 �3 = �4 = 2
y(x) = c1 + c2x+ c3e
2x + c4xe
2x ,
y0(x) = c2 + 2c3e2x + c4e2x[1 + 2x] ,
y00(x) = 4c3e2x + c4e2x[4 + 4x] ,
y(3)(x) = 8c3e
2x + c4e
2x[12 + 8x] .
26664
1 1 e2 e2
0 1 2e2 3e2
0 0 4e2 8e2
0 0 8e2 20e2
37775
0BBB@
c1
c2
c3
c4
1CCCA =
0BBB@
�1
2
0
0
1CCCA .
c1 = �3 c2 = 2 c3 = c4 = 0
y(x) = �3 + 2x , x 2 R .
⇤
8<:y(4) + y0 = 0;y(0) = 0, y0(0) = 0, y00(0) = �1, y(3)(0) = 0.8<:y(4) � y0 = 0;y(0) = 0, y0(0) = 0, y00(0) = 1, y(3)(0) = 1.8<:y(4) + 6y(3) + 17y00 + 22y0 + 14y = 0;y(0) = 1, y0(0) = �2, y00(0) = 0, y(3)(0) = 3.
any
(n) + an�1y(n�1) + · · ·+ a1y0 + a0y = g(x),
a0, a1, ..an g(x)
k, xn, xneax, xneax cos bx, xneax sen bx,
k, a, b n
g(x) = 2x3 + x � 1
g(x) = e2x cos 3x g(x) = sen ⇡x + 5x4 g(x) = x2e�x sen 4x + e4x
g(x)
g
g
g(x) = lnx
g(x) =
1
x
g(x) = tg x
g(x)
y00 + 4y0 � 2y = 7x+ 1.
g(x) = 7x + 1
yp(x) = Ax + B
g yp(x)
4A� 2Ax� 2B = 7x+ 1,
4A�2B = 1
�2A = 7 A = �7/2 B = �15/2
yp(x) = �7
2
x� 15
2 ⇤
y000 � y = x3.
yp = Ax3+Bx2+Cx+D
yp
6A� Ax3 �Bx2 � Cx�D = x3.
⇤
�A = 1 B = 0 C = 0 6A�D = 0 A = �1, B = 0, C = 0
D = �6 yp = �x3 � 6
y00 + 4y = 2 sen 3x
z2 + 4 = 0 �1 = 2i �2 = �2i
yh(x) = c1 cos 2x + c2 sen 2x
yp(x) = A sen 3x+B cos 3x
g(x)
�9A sen 3x� 9A cos 3x+ 4A sen 3x+ 4B cos 3x = 2 sen 3x.
sen 3x cos 3x
�5A = 2 �9A+ 4B = 0 A = �2/5 B = �9/10
y(x) = c1 cos 2x+ c2 sen 2x� 2
5
sen 3x� 9
10
cos 3x.
⇤
y00 � 5y0 + 4y = ex.
�1 = 1 �2 = 4 yh = c1ex + c2e4x
g
yp(x) = Aex
x
yp = Axe
x.
2Aex + Axex � 5Aex � 5Axex + 4Axex = ex,
A = �1/3 yp = �1
3
xex
y(x) = c1ex + c2e4x � 1
3
xex ⇤
y00 � 2y0 + y = 2ex.
�1 = �2 = 1 yh = c1ex + c2xex
yp = Ax
2ex,
y = ex y = xex
x
2Aex = 2ex
A = 1 y(x) = c1ex + c2xex + x2ex ⇤
y00 + y = cos x+ x� 3e2x.
yh(x) = c1 cosx + c2 sen x
g
yp = Ax cosx+Bx sen x+Cx+D +Ee2x
�2A sen x+ 2B cosx+ Cx+D + 5Ee2x = cos x+ x� 3e2x.
�2A = 0, 2B = 1, C = 1, D = 0 5E = �3
yp =
1
2
x sen x+ x� 3
5
e2x
y(x) = c1 cosx+ c2 sen x+
1
2
x sen x+ x� 3
5
e2x ⇤
g1 g2
an(x)y
(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g1(x)
an(x)y
(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g2(x)
y1(x) y2(x) y1(x)+y2(x)
g1 g2
an(x)y
(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g1(x) + g2(x).
g
y000 � y00 + y0 � y = x2ex.
z3�z2+z�1 = 0
z2(z�1)+z�1 = (z�1)(z2+1) = 0
�1 = 1 �2 = �i �3 = i yh(x) = c1ex + c2 cosx + c3 sen x
yp(x) = x(Ax
2 +Bx+ C)ex.
x
A = 1/6
B = �1/2 C = 1/2
y(x) = c1e
x + c2 cosx+ c3 sen x+ (
1
6
x3 � 1
2
x2 +
1
2
x)ex.
⇤
y00 + 3y = 5
2y00 � y0 + 2y = 3xex
y00 � 2y0 + y = 2ex + xex
y000 + 3y00 � 4y = ex cosx
y00 + 2y = xex sen x
y(4) � y = �ex
y00 + 4y = senx cosx sen 2x = 2 senx cosx
y00 + 3y = 5 y0(0) = 1 y(0) = �1
y000 + y00 + 3y0 + 3y = x2 y00(0) = 2 y0(0) = 1, y(0) = 0
n
n
y(n) + an�1y(n�1) + · · ·+ a1y0 + a0y = g,
a0, · · · , an�1 g x
a0, · · · , an g
n = 2
y00 + a1(x)y0 + a0(x)y = g(x),
yh(x) = c1y1(x) + c2y2(x),
c1c2
y00 + a1(x)y0 + a0(x)y = 0.
c1, c2
yp(x) = c1(x)y1(x) + c2(x)y2(x),
c1 c2
y1 y2 g c1
c2
c1(·) c2(·)
y0p(x) = c
0
1(x)y1(x) + c
0
2(x)y2(x) + c1(x)y
0
1(x) + c2(x)y
0
2(x),
c1 c2
c01(x)y1(x) + c
0
2(x)y2(x) = 0.
y0p(x) = c1(x)y
0
1(x) + c2(x)y
0
2(x).
y00p(x) = c
0
1(x)y
0
1(x) + c
0
2(x)y
0
2(x) + c1(x)y
00
1(x) + c2(x)y
00
2(x) .
a0(x)[c1(x)y1(x) + c2(x)y2(x)] + a1(x)[c1(x)y
0
1(x) + c2(x)y
0
2(x)] +
c1(x)y
00
1(x) + c2(x)y
00
2(x) + c
0
1(x)y
0
1(x) + c
0
2(x)y
0
2(x) = g(x).
y1 y2
y001 + a1(x)y
0
1 + a0(x)y1 = 0 y
00
2 + a1(x)y
0
2 + a0(x)y2 = 0,
c1 c2
c01(x)y
0
1(x) + c
0
2(x)y
0
2(x) = g(x).
c1 c2
c01(x)y1(x) + c
0
2(x)y2(x) = 0,
c01(x)y
0
1(x) + c
0
2(x)y
0
2(x) = g(x).
x
c01(x) =
����� 0 y2(x)g(x) y02(x)
����������y1(x) y2(x)y01(x) y02(x)
�����
, c02(x) =
�����y1(x) 0y01(x) g(x)
����������y1(x) y2(x)y01(x) y02(x)
�����
.
W (y1(x), y2(x)) y1
N � 3
y2
x
c1(x) = �
Z
y2(x)g(x)
W (y1(x), y2(x))
dx , c2(x) =
Z
y1(x)g(x)
W (y1(x), y2(x))
dx .
n � 3
n � 3
2
y(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = g(x),
y(x) = c1y1(x) + c2y2(x) + · · ·+ cnyn(x),
c1, c2, · · · , cn
y(n) + an�1(x)y(n�1) + · · ·+ a1(x)y0 + a0(x)y = 0.
yp(x) = c1(x)y1(x) + c2(x)y2(x) + · · ·+ cn(x)yn(x),
c1, c2, · · · , cn
y0p c1, c2, · · · , cn
c01(x)y1(x) + c
0
2(x)y2(x) + · · ·+ c0n(x)yn(x) = 0,
y0p(x) = c1(x)y
0
1(x) + c2(x)y
0
2(x) + · · ·+ cn(x)y0n(x).
y00p c1, · · · , cn
c01(x)y
0
1(x) + c
0
2(x)y
0
2(x) + · · ·+ c0n(x)y0n(x) = 0,
y00p(x) = c1(x)y
00
1(x) + c2(x)y
00
2(x) + · · ·+ cn(x)y00n(x).
c01(x)y
(n�1)
1 (x) + c
0
2(x)y
(n�1)
2 (x) + · · ·+ c0n(x)y(n�1)n (x) = 0,
y(n�1)p (x) = c1(x)y
(n�1)
1 (x) + c2(x)y
(n�1)
2 (x) + · · ·+ cn(x)y(n�1)n (x).
y(n)p
y1, · · · , yn
c1, c2, · · · , cn
c01(x)y
(n�1)
1 (x) + c
0
2(x)y
(n�1)
2 (x) + · · ·+ c0n(x)y(n�1)n (x) = g(x).
c1, c2, · · · , cn
c01(x)y1(x)+ c
0
2(x)y2(x) + · · ·+ c0n(x)yn(x) = 0
c01(x)y
0
1(x)+ c
0
2(x)y
0
2(x) + · · ·+ c0n(x)y0n(x) = 0
= 0
c01(x)y
(n�1)
1 (x)+ c
0
2(x)y
(n�1)
2 (x) + · · ·+ c0n(x)y(n�1)n (x) = g(x) .
N � 3
x
c0i(x) =
(�1)n+ig(x)Mn,i(x)
W (y1(x), y2(x), · · · , yn(x)) 1  i  n .
W (y1(x), y2(x), · · · , yn(x)) y1, · · · , yn
Mn,i(x)
266664
y1(x) y2(x) · · · yn(x)
y01(x) y
0
2(x) · · · y0n(x)
y(n�1)1 (x) y
(n�1)
2 (x) · · · y(n�1)n (x)
377775
n i
x
ci(x) = (�1)n+i
Z
g(x)Mn,i(x)
W (y1(x), y2(x), · · · , yn(x)) dx 1  i  n .
y00 + y = tg t, 0 < t <
⇡
2
.
yh = c1 cos t + c2 sen t y00 + y = 0
yp
yp(t) = c1(t) cos t+ c2(t) sen t,
c1(t) = �
Z
sen t tg t
W (cos t, sen t)
dt c2(t) =
Z
cos t tg t
W (cos t, sen t)
dt ,
W (cos t, sen t) =
����� cos t sen t� sen t cos t
����� = 1.
c1(t) = �
Z
sen t tg tdt = tg t cos t�
Z
cos t sec2 tdt = sen t�ln(sec t+tg t)+C
c2(t) =
Z
cos t tg tdt =
Z
sen tdt = � cos t+D.
C = D = 0
c1(t) c2(t)
yp(t) = [sen t� ln(sec t+ tg t)] cos t� cos t sen t = � ln(sec t+ tg t) cos t
y(t) = c1 cos t+ c2 sen t� ln(sec t+ tg t) cos t .
⇤
y00 � 2y0 + y = e
t
1 + t2
.
yh = c1et + c2tet y00 � 2y0 + y = 0
yp
yp(t) = c1(t)e
t + c2(t)te
t,
c1(t) = �
Z
te2t
e2t(1 + t2)
dt c2(t) =
Z
e2t
e2t(1 + t2)
dt ,
W (et, tet) =
�����et tetet et(1 + t)
����� = e2t.
c1(t) = �
Z
t
1 + t2
dt =
1
2
ln(1 + t2) + C
c2(t) =
Z
1
1 + t2
dt = arctg t+D.
C = D = 0 c1(t) c2(t)
y(t) = c1e
t + c2te
t +
1
2
ln(1 + t2)et + arctg(t) tet .
⇤
y000 � y00 + y0 � y = sec t, �⇡
2
< t <
⇡
2
.
yh = c1et + c2 cos t+ c3 sen t
y000 � y00 + y0 � y = 0
yp
yp(t) = c1(t)e
t + c2(t) cos t+ c3(t) sen t,
c1(t) =
1
2
Z
e�t sec t dt, c2(t) = �1
2
Z
sec t(cos t� sen t) dt
c3(t) = �1
2
Z
sec t(cos t+ sen t) dt,
W (et, cos t, sen t) =
�������
et cos t sen t
et � sen t cos t
et � cos t � sen t
������� = 2et,
M3,1(t) =
����� cos t sen t� sen t cos t
����� = 1, M3,2(t) =
�����et sen tet cos t
����� = et(cos t� sen t),
M3,3(t) =
�����et cos tet � sen t
����� = �et(cos t+ sen t).
c2(t) = �1
2
[t+ ln(cos t)] + C
c3(t) =
1
2
[ln(cos t)� t] +D.
h(x) = e�t sec t
h C = D = 0 c1(t) c2(t) c3(t)
y(t) = c1e
t+c2 cos t+ c3 sen t+
⇣Z t
0
e�s sec s ds
⌘
et
�1
2
[t+ ln(cos t)] cos t+
1
2
[ln(cos t)� t] sen t .
⇤
y00 + 9y = sec2(3t) 0 < t < ⇡6
y00 + 4y = 3 csc(2t) 0 < t < ⇡2
y000 + y = sec t �⇡2 < t < ⇡2
y000 � y = csc t 0 < t < ⇡
x2y00 � 2y = 3x2 � 1 x > 0 y1(x) = x2 y2(x) = x�1
x3y000 + x2y00 � 2xy0 + 2y = 2x4 x > 0 y1(x) = x2 y2(x) = x�1
ax2y00 + bxy0 + cy = 0, x 6= 0.
x > 0
x
y = xm.
ax2m(m� 1)x2xm�2 + bmxxm�1 + cxm = 0, x > 0,
am(m� 1)xm + bmxm + cxm = 0, x > 0.
y = xm m
am(m� 1) + bm+ c = 0.
am2 + (b� a)m+ c = 0
� = (b� a)2 � 4ac
•
m1 m2
y1(x) = xm1
y2(x) = xm2
y(x) = c1x
m1 + c2x
m2 .
•
m =
a� b
2a
y1(x) = x
a�b
2a
y2
y2(x) = x
a�b
2a
Z
e�
R b
axdx
x
a�b
a
dx = x
a�b
2a
Z
1
x
dx = x
a�b
2a lnx, x > 0.
y(x) = c1x
a�b
2a + c2x
a�b
2a lnx, x > 0.
•
m1 = ↵+ i�
m2 = ↵� i�
z1(x) = x
↵+i�, z2(x) = x
↵�i�.
z1(x) = x
↵+i� = x↵xi� = x↵ei� lnx = x↵ cos(� lnx) + ix↵ sen(� lnx),
z2(x) = x
↵ cos(� lnx)� ix↵ sen(� lnx).
y1 = x↵ cos(� lnx) y2 = x↵ sen(� lnx)
y(x) = c1x
↵ cos(� lnx) + c2x
↵ sen(� lnx).
x2y00 � 2xy0 � 4y = 0, x > 0.
y = xm
m2 � 3m� 4 = 0,
m1 = �1 m2 = 4
y(x) = c1x�1 + c2x4 ⇤
4x2y00 + 8xy0 + y = 0, x > 0.
y = xm
4m2 + 4m+ 1 = 0,
m = �1/2
y(x) = c1x�1/2+c2x�1/2 lnx
⇤
x2y00 + 3xy0 + 3y = 0, x > 0.
m2 + 2m+ 3 = 0,
m1 = �1 + i
p
2 m2 = �1� i
p
2
y(x) = c1x�1 cos(
p
2 ln x) + c2x�1 sen(
p
2 ln x) ⇤
x2y00 + 3xy0 + y = ln x, x > 0.
m = �1
yh(x) = c1x�1 + c2x�1 lnx
yp(x) = c1(x)x
�1 + c2(x)x�1 lnx,
c1(x) = �
Z
x�2 lnx.x�1 lnx
W (x�1 lnx, x�1)
dx , c2(x) =
Z
x�2 lnx .x�1
W (x�1 lnx, x�1)
dx .
c1(x) c2(x)
c1(x) = �x ln2 x+ 2x lnx� 2x e c2(x) = x lnx� x.
y(x) = c1x
�1 + c2x�1 lnx+ lnx� 2, x > 0.
N � 3
⇤
y(x) x > 0
y(�x) x < 0
x y(|x|)
x
x 6= 0 y(x) = c1|x|�1 cos(
p
2 ln |x|) + c2|x|�1 sen(
p
2 ln |x|)
y(x) = c1x�1 cos(
p
2 ln |x|) + c2x�1 sen(
p
2 ln |x|)
n � 3
n
anx
ny(n) + an�1xn�1y(n�1) + ...+ a1xy0 + a0y = 0, x > 0.
y = xm
x = et
x > 0
t
x = et
n = 3
a3x
3 d
3y
dx3
+ a2x
2 d
2y
dx2
+ a1x
dy
dx
+ a0y = 0, x > 0.
x = et t = lnx
dt
dx
=
1
x
y x
y t
dy
dx
=
dy
dt
dt
dx
=
dy
dt
1
x
.
d2y
dx2
=
d2y
dt2
1
x2
� dy
dt
1
x2
=
1
x2

d2y
dt2
� dy
dt
�
.
d3y
dx3
=
1
x2

d3y
dt3
1
x
� d
2y
dt2
1
x
�
� 2
x3

d2y
dt2
� dy
dt
�
=
1
x3

d3y
dt3
� 3d
2y
dt2
+ 2
dy
dt
�
.
t
a3
d3y
dt3
+ (a2 � 3a3)d
2y
dt2
+ (2a3 � a2 + a1)dy
dt
+ a0y = 0, t 2 R.
x t = lnx
x3y000 + 6x2y00 + 7xy0 + y = 0, x > 0.
m = �1 x = et
N � 3
a3 = 1, a2 = 6, a1 = 7 a0 = 1 t
y000(t) + 3y00(t) + 3y0(t) + y(t) = 0, t 2 R.
z3 + 3z2 + 3z + z = 0
� = �1
y(t) = c1e
�t + c2te�t + c3t2e�t, t 2 R.
x t = ln x
y(x) =
c1
x
+
c2
x
lnx+
c3
x
ln2 x.
x > 0 ⇤
x3y000 + 3x2y00 + xy0 � y = ln x, x > 0.
x = et
t
y000 � y = t
z3 � 1 = 0
�1 = 1 �2 =
�1 + ip3
2
�3 =
�1� ip3
2
yh(t) = c1e
t + c2e
�t/2 cos(
p
3
2
t) + c3e
�t/2 sen(
p
3
2
t), t 2 R.
yp(t) = At+B
A = �1 B = 0
y(t) = c1e