Formulário

Prévia do material em texto

FUNÇÃO DERIVADA DA FUNÇÃO 
y(x) = C y’(x) = 0 
y(x) = C.u(x) y’(x) = C.u’(x) 
y(x) = xn y’(x) = nxn-1 
y(x) = u(x) ± v(x) y’(x) = u’(x) ± v’(x) 
y(x) = u(x).v(x) y’(x) = u’(x).v(x) + u(x).v’(x) 
y(x) = u x
v x
( )
( ) y’(x) = u x v x u x v xv x' ( ) . ( ) ( ) . ' ( )( )− 2
 
 
y = [u(x)]n y’ = n[u(x)]n-1.u’(x) 
 
y = sen u(x) y’ = cos u(x).u’(x) 
y = cos u(x) y’ = -sen u(x) .u’(x) 
y = tg u(x) y’ = sec2 u(x).u’(x) 
y = cotg u(x) y’ = -cossec2 u(x).u’(x) 
y = sec u(x) y’ = sec u(x).tg u(x).u’(x) 
y = cossec u(x) y’ = -cossec u(x).cotg u(x).u’(x) 
 
y = arc sen u(x) y’ = u x
u
'( )
1 2−
 
 
y = arc cos u(x) y’ = −
−
u x
u
'( )
1 2
 
 
y = arc tg u(x) y’ = u x
u
'( )
1 2+
 
 
y = arc cotg u(x) y’ = −
+
u x
u
' ( )
1 2
 
 
y = arc sec u(x) y’ = u x
u u
'( )
2 1−
 
 
y = arc cossec (x) y’ = −
−
u x
u u
'( )
2 1
 
 
y = eu(x) y’ = eu(x).u’(x) 
 
y = au(x) y’ = au(x).ln a . u’(x) 
 
y = ln u(x) y’ = u x
u x
' ( )
( )
 
 
y = loga u(x) y’ = u x
u x a
'( )
( ).ln
 
y = senh u(x) y’ = cosh u(x).u’(x) 
y = cosh u(x) y’ = senh u(x).u’(x) 
y = tgh u(x) y’ = sech2 u(x).u’(x) 
y = cotgh u (x) y’ = -cossech2 u(x).u’(x) 
y = sech u(x) y’ = -sech u(x).tgh u(x).u’(x) 
y = cossech u(x) y’ = -cossech u(x).cotgh u(x).u’(x) 
 
y= arcsenh u(x) y’ = 
1
)('
2 +u
xu
 
y= arctgh u(x) y’ = 
21
)('
u
xu
−
 
 
 
ÁREAS 
Se f(x) > 0 para x ∈ (a , b) 
A f x dx
a
b
= ∫ ( ) 
Se f2(x) > f1(x) para x ∈ (a , b) 
∫ −=
b
a
12 dx)x(f)x(fA 
 
FÓRMULAS DE INTEGRAÇÃO 
x dx
x
r
Cr
r
=
+
+
+
∫
1
1 com r ≠ -1 
dx x C= +∫ 
e dx e Cx x= +∫ 
e dx e Cx x− −= − +∫ 
1
x
dx x C= +∫ ln 
a
a
a
Cx
x
= +∫ ln
 
sen cosx dx x C= − +∫ 
cos senx dx x C= +∫ 
tgx dx x C= − +∫ ln cos 
cot ln sengx dx x C= +∫ 
sec ln secx dx x tgx C= + +∫ 
cossec ln cossec cotx dx x gx C= − +∫ 
1
2
2
cos
sec
x
dx x dx tgx C= = +∫∫ 
1
2
2
sen
cossec cot
x
dx x dx gx C= = − +∫∫ 
sec . secx tgx dx x C= +∫ 
cossec .cot cos secx gx dx x C= − +∫ 
dx
x a a
x
a
C2 2
1
+
= +∫ arctg 
dx
a x
x
a
C
2 2
−
= +∫ arcsen 
dx
x x a a
x
a
C
2 2
1
−
= +∫ arc sec 
x a dx x x a a x x a C2 2 2 2
2
2 2
2 2
± = ± ± + ± +∫ ln 
sec sec . ln sec3 1
2
1
2
x x tgx x tgx C= + + +∫ 
∫
∫
∫
∫
∫
∫
C +u cosech - =du u cotgh u echcos
C +u sech - =du u u tgh hsec
C +cotghu - =du u echcos
C +u tgh =du u hsec
C +u senh =du u cosh
C +u cosh =du u senh
2
2
 
INTEGRAL POR PARTES 
 
∫ ∫−= duvv.udvu 
 
DISTÂNCIA ENTRE DOIS PONTOS 
 
( ) ( )d x x y yAB A B A B= − −2 2 + 
 
 
 
 
 
 
VOLUMES 
 
[ ]
[ ]
( ) ( )[ ]
V f x dx
V f x k dx
V f x f x dx
a
b
a
b
a
b
=
= −
= −
∫
∫
∫
pi
pi
pi
( )
( )
( ) ( )
2
2
2
2
1
2
 
V r
V r h
V r h
e s f e r a
c i l i n d r o
c o n e
=
=
=
4
3
1
3
3
2
2
pi
pi
pi
.
. .
. .
 
 
PRODUTOS DE SENOS E COSSENOS 
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
sen .cos sen
sen .cos sen sen
cos .cos cos cos
cos .sen sen sen
sen .sen cos cos
u u u
u v u v u v
u v u v u v
u v u v u v
u v u v u v
=
= + + −
= + + −
= + − −
= − − +
1
2
2
1
2
1
2
1
2
1
2
 
 
ARCOS DUPLOS 
 
sen .sen .cos
cos cos sen
cos .cos
cos sen
.
2 2
2
2 2 1
2 1 2
2
2
1
2 2
2
2
2
x x x
x x x
x x
x x
tg x
tg x
tg x
=
= −
= −
= −
=
−
 
 
ADIÇÃO DE ARCOS 
( )sen sen .cos sen .cos
cos( ) cos .cos sen .sen
cos( ) cos .cos sen .sen
sen( ) sen .cos sen .cos
a b a b b a
a b a b a b
a b a b a b
a b a b b a
+ = +
+ = −
− = +
− = −
 
 
EQUAÇÕES EXPONENCIAIS 
a 
n 
 = a 
m
 � n = m 
a 
n + m
 � a 
n
 . a 
 m
 
a 
m - n
 � a 
m
 ÷ a
 n
 
a 
- n
 � 1/ a n 
 
SUBSTITUIÇÃO TRIGONOMÉTRICA 
 
a u2 2−
a
u
a u2 2−
a
u
a u2 2+
a u2 2+
a
u
u a2 2− u a2 2−
 
 
 
RELAÇÕES TRIGONOMÉTRICAS NO TRIÂNGULO RETÂNGULO 
 
sen
sec
θ
θ
θ
=
=
=
CO
H
tg
CO
CA
H
CA
cos
cot
cossec
θ
θ
θ
=
=
=
CA
H
g
CA
CO
H
CO
H
CO
CA
0 .
 
 
 
IDENTIDADES TRIGONOMÉTRICAS 
sen cos2 2 1x x+ = - Identidade fundamental da trigonometria 
 
tg x x
x
x
x
x
g x
tg x
x
x
=
=
=
= =
sen
co s
sec
co s
co s sec
sen
co t
co s
sen
1
1
1
 x
 
 
xgx
xxg
xtgx
22
22
22
cot1seccos
1seccoscot
1sec
+=
−=
=−
 
sen
cos
cos
cos
2
2
1 2
2
1 2
2
x
x
x
x
=
−
=
+
 
LOGARITMOS 
 
log b x = y se b y = x 
log e x = x se e ln x = x 
log b MN = log bM + log b N 
log b 
M
N
= log bM - log b N 
log b 1 = 0 
log a M n = n log a M 
log a a = 1 
log b b n = n ou ln e n = n