Gabarito - Lista de Integral Indefinida

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ECT1102 - Cál
ulo I
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Gabarito - 3
a
Lista - Primitivas e Integração:
1.
(a) F (x) = 3x3 − 2x2 + 3x + C (b) F (x) = 1
2
x4 − 1
3
x3 +
3
2
x2 + C (c) F (x) = − 1
2x2
+
3
x
+ Cx + K
(d) F (x) = 2
√
x3 + 2
√
x + C (e) F (x) = 9
3
√
x2 − 1
8
3
√
x4 + 7x + C (f) F (x) =
8
9
x
9
4 +
24
5
x
5
4 − x−3 + C
(g) F (x) =
1
4
x8 − ex + C (h) F (x) = − cosx + 2
3
x
3
2 + C (i) F (x) = senx +
3
2
x−
4
3
2.
(a)
∫
(x + 1)dx =
x2
2
+ x + C (b)
∫ (
3t2 +
t
2
)
dt = t3 +
t2
4
+ C
(c)
∫
x−
1
3 dx =
3
2
x
2
3 + C (d)
∫
t
√
t +
√
t
t2
dt =
2t− 2√
t
+ C
(e)
∫ (
e−x + 4x
)
dx = −e−x + 4
x
ln 4
+ C (f)
∫
(4 se
x tgx− 2 se
2x)dx = 4 se
x− 2 tgx+ C
(g)
∫
7 sen
(
θ
3
)
dθ = −21 cos θ
3
+ C (h)
∫ [
sen(2x)− 
osse
2(x)] dx = − 1
2
cos(2x) + 
otg(x) + C
(i)
∫
x3exdx = x3ex − 3x2ex + 6xex − 6ex + C
3.
(a) y(x) = x− 1
x
(b) y(x) =
x3
3
+ 2x− 1
x
− 1
3
(c) y(x) = −e−x + 11
(d) y(x) = senx− cosx + 3 (e) r(t) = 4
√
t5 + 4
√
t3 − 8t− 16 (f) r(t) = sent− t− 1
4. y(x) = 2
√
x3 − 50.
5. V (t) = 2
√
t3 + 1
8
t2 + 2.
6. v = 1200m/s.
7.
(a)
∫ 4
1
(x2 − 4x− 3)dx = −18 (b)
∫ 3
−2
(8x3 + 3x− 1)dx = 265
2
(c)
∫ 12
7
dx = 5
(d)
∫ 9
4
x− 3√
x
dx =
20
3
(e)
∫ 3
−2
|x|dx = 13
2
(f)
∫ 2
3
(
x2 − 1
x− 1
)
dx = −7
2
(g)
∫ 6
−3
|x− 4|dx = 8 (h)
∫ 4
0
√
3x
(√
x+
√
3
)
dx = 8
√
3 + 16 (i)
∫ 4
0
x√
x2 + 9
dx = 2
(j)
∫ 0
−2
3
√
x + 1 dx = 0 (k)
∫ 5
0
√
x + 4 dx =
38
3
(l)
∫ 2
−3
√
6− x dx = 38
3
(m)
∫ 1
0
e−xdx =
e− 1
e
(n)
∫ 0
−pi/2
cosxdx = 1 (o)
∫ pi/2
−pi/2
(1 − cosx)dx = pi − 2
(p)
∫ 0
−∞
ex dx = 1 (q)
∫ 2
0
x ex
2
dx =
e4 − 1
2
(r)
∫ √pi
0
x sen x2 dx = 1
(s)
∫ pi
0
sen2 x dx =
pi
2
(t)
∫ pi
0
x cos2x dx =
pi2
4
(u)
∫ 2pi
0
sen2 x dx = pi
(v)
∫ 2
1
x lnxdx =
8 ln 2− 3
4
(w)
∫ pi/2
0
θ2 sen(2θ) dθ =
pi2 − 4
8
(x)
∫ 2
2/
√
3
t sec−1 t dt =
6 ar
se
(2)− ar
se
(
2
√
3
3
)
− 2√3
3
8.
(a)
√
1− x2
(b)− 1
2
x−
1
2 senx,
(d)1, .
9.
(d)51
4
; .
10.
(a)32
3
; (b)48
5
;
(c)8; (d)243
8
;
(e)8
3
; (f)104
15
;
(g)56
15
; (h)4;
(i)4
3
− 4pi ;
11.
(a)− 1
3
(3− 2s) 32 + C, (b)(x2 − 7x + 7) ex + C,
(d)− 2x cos(x
2
) + 4 sen(x
2
) + C, (e) 2 (senv)
3
2 + C ,
(f)2 ln(
√
x + 1) + C, (g)− ln |cosec(s− pi) + cotg(s− pi)|+ C,
(h)3
x+1
ln 3
+ C, (k)tgx− 2 ln |cosecx + cotgx| − cotgx− x+ C,
(l)tgx− secx + C (n)2
3
(
√
3s + 9e
√
3s+9 − e
√
3s+9) + C
(o)sen−1x +
√
1− x2 + C, (q)x− ln |x + 1|+ C,
(r)x tgx + ln |cosx|+ C, (s)1
2
(−eθ cosθ + eθsenθ) + C,
(t)t2 sen(t) + 2t cos(t)− 2 sen(t) + C, (v) ln |1 + senθ|+ C,
(w)1
2
[−x cos(ln x) + xsen(lnx)] + C, (x) e2x
13
(3sen(3x) + 2cos(3x)) + C, .