lista12_cal1

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Lista de Exerc´ıcios 12
1. Use uma substituic¸a˜o trigonome´trica
apropriada e calcule as seguintes inte-
grais indefinidas.
(a)
∫ √
4− x2dx
(b)
∫
x
2
(4− x2)3/2
dx
(c)
∫
dx
x2
√
4− x2
(d)
∫ √
9− x2
x2
dx
(e)
∫
dx
(
√
x2 + 3)3
(f)
∫
dx
x2
√
x2 + 9
(g)
∫
dx
x
√
x2 + 5
(h)
∫
1
x2 + a2
dx
(i)
∫
dx
x3
√
x2 − 25
(j)
∫
dx
x2 − 16
(k)
∫
dx
9x2 − 1
(l)
∫
dx
x3
√
x2 − 16
2. Complete os quadrados, e usando uma
substuic¸a˜o trigonome´trica, calcule as
seguintes integrais:
(a)
∫
dx
(5− 4x− x2)3/2
(b)
∫
xdx
√
4x2 + 8x+ 5
(c)
∫
x
x2 − 4x+ 8
dx
(d)
∫
dx
5− 4x+ 2x2
3. Use frac¸o˜es parciais e calcule as
seguintes integrais indefinidas.
(a)
∫
3x− 5
x2 − x− 2
dx
(b)
∫
5x3 − 6x2 − 68x− 16
x3 − 2x2 − 8x
dx
(c)
∫
3x2 + 4x+ 2
x(x+ 1)2
dx
(d)
∫
3x− 2
x3 − x2
dx
(e)
∫
8x2 + 3x+ 20
(x+ 1)(x2 + 4)
dx
(f)
∫
3x3 + 2x2 − 2
x2(x2 + 2)
dx
(g)
∫
x
3 + x+ 2
x(x2 + 1)2
dx
(h)
∫
x
5 − 2x4 + 2x3 + x− 2
x2(x2 + 1)2
dx
4. Use a identidade
sen2x+ cos2x = 1,
e substituic¸o˜es apropriadas para calcular
cada integral.
(a)
∫
cos
3
x dx
(b)
∫
sen
5(2x)dx
(c)
∫
sen
2
cos
5
x dx
(d)
∫
sen
3
x
√
cos x
dx
1
5. Use as identidades trigonome´tricas
sen2x =
1
2
(1− cos 2x), cos2x = 1
2
(1 + cos 2x),
e substituic¸o˜es apropriadas para calcular
cada integral.
(a)
∫
cos
2
x dx
(b)
∫
sen
4
x dx
(c)
∫
sen
4
x cos
4
x dx
(d)
∫
sen
4
x cos
2
x dx
6. Use as identidades trigonome´tricas
sen x cos y = 1
2
[sen(x+ y) + sen(x− y)] ,
sen x sen y = 1
2
[cos(x− y)− cos(x+ y)] ,
cos x cos y = 1
2
[cos(x− y) + cos(x+ y)] ,
e substituic¸o˜es apropriadas para calcular
cada integral.
(a)
∫
(sen 3x)(cos 4x)dx
(b)
∫
(sen 7x)(cos 2x)dx
(c)
∫
(sen 5x)(cos 2x)dx
(d)
∫
(cos4x)(cos3x)dx
Respostas
1. (a) 2arcsen x+ x
2
√
4− x2 + C
(b) x√
4−x2
− arcsen (x
2
)
+ C
(c) −
√
4−x2
x
+ C
(d) − 3
x
− arcsen (x
3
)
+ C
(e) x
3
√
3+x2
+ C
(f) −
√
x2+9
9x
+ C
(g) 1√
5
ln
∣∣∣∣
√
x2+5−√5
x
∣∣∣∣+ C
(h) 1
a
arctg
(
x
a
)
+ C
(i) 1
250
(
arcsen
(
x
5
)− 5
√
x2−25
x2
)
+ C
(j) ln
∣∣∣∣x4 +
√
x2+16
4
∣∣∣∣+ C
(k) 1
6
ln
∣∣∣ 3x−13x+1
∣∣∣+ C
(l) 1
128
(
arcsen
(
x
4
)− 4
√
x2−16
x2
)
+ C
2. (a) x+2
9
√
9−(x+2)2 + C
(b) 1
4
√
4x2 + 8x+ 5− 1
2
ln |√4x2 + 8x+ 5+2x+2|+
C
(c) 1
2
ln
(
(x− 2)2 + 4)+ arctg (x−2
2
)
+ C
(d) 1√
2
arcsen
(√
2/7(x+ 1)
)
+ C
3. (a) ln(|x− 2|1/3|x+ 1|8/3) + C
(b) 5x+ ln
(
x2|x+2|14/3
|x−4|8/3
)
+ C
(c) ln |x2(x+ 1)|+ 1
x+1
+ C
(d) − ln |x| − 2
x
+ ln |x− 1|+ C
(e) 3
2
ln(x2 + 4) + 5 ln |x+ 1|+ C
(f) ln |x|+ 1
x
+ ln(x2 + 2) +
√
2
2
arctg
√
2
2
x+ C
(g) 2 ln |x| − ln(x2 + 1) + arctg x+ 1
x2+1
+ C
(h) 1
2
arctg x+ x
2(x2+1)
+ C
4. (a) sen x− sen3x
3
+ C
(b) − 1
2
(
cos 2x− 2
3
cos3(2x) + 1
5
cos5(2x)
)
+ C
(c) sen
3x
3
− 2sen5x
5
+ sen
7x
7
+ C
(d)
5(cos x)5/2
2
− 2√cos x+ C
5. (a) x
2
+
sen(2x)
4
+ C
(b) 3x
8
− sen 2x
4
+ sen 4x
32
+ C
(c) 1
32
(
3x
4
− sen4x
4
+ sen 8x
32
)
+ C
(d) 1
8
(
x
2
− sen 4x
8
− sen32x
6
)
+ C
6. (a) − cos7x
14
+ cos x
2
+ C
(b) − 1
10
cos 9x− 1
10
cos 5x+ C
(c) − 1
14
cos 7x− 1
6
cos 3x+ C
(d) 1
2
sen x+ 1
14
sen 7x+ C
2