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LISTA DE EXERCÍCIOS
1. CALCULE:
a. ∫
1 2 1 + t 2 t 4 dt SOLUÇÃO
1 2 1 + t 2 t 4 dt
1 + t 2 t
4 dt =∫^
t
4 +^
t 2 t
4 dt =∫ t
− 4
- t 2 t − 4 dt = t − 3 − 3
- t − 2 =¿ ¿ ¿−
3 t
3 +^
t − 1 − 1
3 t 3 +(
t )
3 t
3 −^
t
1 2 − 1 3 t
3 −^
t dt = [
3 t
3 −^
t ] 1 2 =− [
3 t (^3) ] 1 2 −[
t ] 1 2 =¿ ¿− [
3 × 2
3 −^
3 × 1
(^3) ]−[
1 ]
[
3 ]
[
2 ]
[
24 ]
[
2 ]
[
24 ]
1 2 1 + t 2 t 4 dt =^
b. ∫
− 1 1 x 7
− 1 1 x 7
∫(^ x
7
+ x )^ dx =¿
x 8
x 4
x 2
− 1 1
x 8 8
x 4 4
x 2
dx =[
x 8 8
x 4 4
x 2
2 ]− 1
1
8 8
4 4
2 2
8 8
4 4
2 2
− 1 1 x 7
c. ∫
0 π 2
cos 2 x ) dx
SOLUÇÃO
x +
×
sen 2 x dx =∫
x +
sen 2 x + k
0 π 2 1 2
sen 2 x dx =¿ [
x +
sen 2 x ]
0 π (^2) =¿ ¿ ¿
× (
π
sen 2 π 2
× ( 0 )+
sen 2 × 0 = π 4
0 π 2
cos 2 x ) dx =
π 4
d. ∫
0 1 3 x e x dx SOLUÇÃO
∫(^3 e^ )^
x dx =¿ ( 3 e ) x ln ( 3 e )
0 1 ( 3 e ) x ln ( 3 e ) dx =
[
( 3 e ) x
ln( 3 e ) ] 0
1
( 3 e ) 1 1 +ln ( 3 )
( 3 e ) 0 0 +ln ( 3 )
3 e − 1 1 + ln ( 3 )
1 3 x e x dx =¿ 3 e − 1 1 + ln( 3 )
[ e 3. 9
e 3. 9 ] −[
cos 3.1−
cos 3.2]=[ e^3 9
e^6 9 ]
[ e 3 − e 6 ]−[ 3 cos ( 3 )− 3 cos ( 6 )]=
[ e 3 − e 6 − 3 cos ( 3 ) + 3 cos ( 6 )]
2 1 (
e 3 x
[ e 3 − e 6 − 3 cos ( 3 ) + 3 cos ( 6 )] ¿
h. ∫
0 π 2 sen 2 x (^) √ 5 + sen^2 x dx SOLUÇÃO
∫ sen^^2 x^ √^5 + sen
2 x u = sen 2 x du = sen 2 x dx
¿∫√ u + 5 du
∫ √ u +^5 du
∫ √ u +^5
s = u + 5 ds = du
¿∫√ s ds =
2 s 3 2 3
( (^) u + 5 ) 3 2 =
( sen^2 x + 5 )
3 2
0 π 2 sen 2 x (^) √ 5 + sen 2
x dx =∫
0 π 2 2 3
( sen
2
x + 5 )
3 2 dx = [
( sen
2
x + 5 )
3 2
i. ∫
0 1 2 x √^1 − x 2 dx
SOLUÇÃO
u = 1 − x 2 ⟹ du = 2 xdx ⟹ x dx = du 2
0 1 2 x √ 1 − x 2 dx =
1 3 4 1 √ u^ du =
3 4 1 u − 1 (^2) du = 1 2
[ 2 u
1
2 ]
3 4 1 =[ (^) √ u ] (^3) 4 1 =¿ ¿ ¿ (^) √ 1 −
√^3 √ 4
√^3 2 j. (^) ∫ x 3 e x^2 dx SOLUÇÃO u = x 2 dv = x e x^2 dx du = 2 x dx v = e x^2 2
∫ x
3 e x^2 dx = x 2 e x^2
−∫ x e
x^2 dx = x 2 e x^2 − e x^2
