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7.6 – EXERCÍCIOS – pg. 325
Nos exercícios de 1 a 23, calcular a integral indefinida.
- dx x x
x
2
3 2
x x x C
x x C
x
dx x
x
x
xdx dx x x
x
2 2 ln 1
ln 1 2
2
2
3 2
- dx x x
x
2
x x C
x x C
dx x x
dx x
B
x
A
dx
x x
x dx
x x
x
ln 2 5
ln 5
ln 2 5
ln 10
2
- dx x x x
x
3 2
dx x
C
x
B
x
A
dx x x x
x
Cálculo de A, B e C
( x − 1 ) ≡ A ( x + 1 ) ( x + 2 ) + B ( x − 2 ) ( x + 2 ) + C ( x − 2 )( x + 1 )
A
x A
=^ −
C
x C
B
x B
Assim,
x x x C
dx x x x
I
ln 2 4
ln 1 3
ln 2 12
- dx x x x
x
3 2
2
dx
x
C
x
B
x
A
x x x
xdx
3 2
2
Cálculo de A, B e C
2 + − +
x ≡ Ax + x − Bx x C x x
1 ( 8 21 18 ) ( 6 9 ) ( 7 16 12 ) ( 4 4 )
3 2 2 3 2 2
2 2 2 2
x A x x x B x x C x x x Dx x
x Ax x Bx Cx x D x
A B C D
A B C D
A B C D
A C
A = 3 ; B = 1 ; C =− 3 e D = 2
Assim,
C
x x x
x
C
x
x x
x
C
x x
x x
dx x x x x
I
− −
3 ln
3 ln 3 2
3 ln 2
3 ln 3 2. 1
3 ln 2
1 1
2 2
( ) dx x x x x
x
4 3 2
2
dx x
D
x
C
x
B
x
A
I 2 3
Cálculo de A, B, C e D
2 3 2 x + ≡ A x − + Bx − x − + C x − x − + D x −
A =− 2 ; B = 2 ; C =− 1 ; D = 5
Assim,
C
x x x
x
C
x x x
x
C
x x
x x
C
x x
I x x
2
2
2
2
2
ln
2 ln
2 ln 1 2 ln 2
2 ln 1 2 ln 2
3 2 x 4 x
dx
dx x
C
x
B
x
A
2
Cálculo de A, B e C
2 1 ≡ Ax x − 4 + Bx − 4 + Cx
x = 0 → 1 =− 4 B ∴ B =−^1
x = 4 → 1 = 16 C ∴ C =^1
A + C = 0 ∴ A =−^1
Assim,
ln 16
ln 4 16
ln 16
ln 4 16
ln 16
1
2
C
x x
x
x C x
x
x C
x x
dx x x x
I
−
- dx x
x x
2
3 2
C
x x x arctg
C
x x x arctg
C
x x x arctg
x x
dx x x
x x
dx dx x x
x
x x
dx dx x x
x
x x
dx
x x
xdx
ln 1 2
ln 1 2
ln 1 2
ln 1 2
2
2
2
2
2
2 2
2 2
2 2
3 x
dx
( ) ( )
dx x x
Bx C
x
A
x x x
dx
2
2
Cálculo de A, B e C
1 ( 2 4 ) ( ) ( 2 )
2 ≡ A x − x + + Bx + C x +
A C
A B C
A B
A = ;
B = e 3
C =
Assim,
C
x x x x arctg
x x
dx
x x
x dx x
x x
dx dx x x
x x
dx x x
dx x x
x x
dx x x
x
x
dx I
ln 2 4 24
ln 2 12
ln 2 12
ln 2 12
ln 2 12
2
2 2
2 2
2 2
2
dx x x
x
2 2 2 3
( ) [( ) ]
C
x arctg x x
x
x x
dx
x x
x
x x
x x
dx
x x
x x x
x
dx
x x
x
x x
dx
x x
x
−
2
2 2 2
2 2
2 1
2 2 2 2
2 2 2 2
2 2 xx x 1
dx
dx x x
Dx E
x x
Bx C
x
A
(^22) (^11)
Cálculo de A, B, C, D e E.
1 ≡ A ( x − x + 1 ) +( Bx + C ) x ( x − x + 1 ) +( Dx + E ) x
2 2 2
x = 0 → 1 = A
≡ A ( x − x + x − x + ) +( Bx + C ) ( x − x + x ) + Dx + Ex
4 3 2 3 2 2 1 2 3 2 1
A B C D
A C D
A B C D
A B C
A =^4 ; B =− 4 ;
C =^68 ;
D =^16
Assim,
C
x x x x x
x
dx
x
dx
x
dx
x
dx I x
−
ln 2 9
ln 1 4 ln 2 9
1
2
dx
x x
x
2
2
x x x C
dx C x x
x
dx
x
B
x
A
dx
dx
x x
x
x x
xdx dx
x x
x
I
ln 45
ln 10
2
2
2
2
2
x + x
dx
9
2
x^ (^ x )^ C
x x C
dx x
x dx x
dx x
Bx C
x
A
^ +
ln 9 2
ln 9
ln 9 2
ln 9
2
2
2
2
( ) ( )
2 2 x x
dx
dx x
Cx D dx x
Ax B
2 2
Calculando A,B,C, e D:
1 ( ) ( 4 ) ( ) ( 1 )
2 2 ≡ Ax + B x + + Cx + D x +
B D
A C
B D
A C
A = 0 ;
B =^1 ; C = 0 ;
D =−^1
Assim,
C
x arctgx arctg
C
x arctgx arctg
dx x
dx x
I
2 2
- dx x
x x x
3
3 2
dx x x
Bx C
x
A
dx x
x x dx x
x x x
2
3
2
3
3 2
( )
( )
C
x
x
C
x x
dx x
x dx x
x I
−
ln 2 2
ln 2 2
2
2
2 1 2
2 2 2
x − x + x − x
dx 4 3 2 3 3
4 3 2 2 3 (^33111)
− + − x
D
x
C
x
B
x
A
x x x x
Calculando A,B,C e D
1 ≡ A ( x − 1 ) + Bx ( x − 1 ) + Cx ( x − 1 ) + Dx
3 2
x D D
x A A
B A B
A B
3 A + B − C + D = 0 ∴ C =− 1
Assim,
C
x x x
x
C
x x x x
dx x x x x
I
− −
2
1 2
2 3
ln
ln ln 1
2 2 x 1 x 1
xdx
2 2 2 1 1 1 1 1 + 1
− + x
D
x
C
x
B
x
A
x x
x
Calculando A,B,C e D
x A ( x x x x x ) Bx Bx B C ( x x x x x ) Dx Dx D
x A x x Bx Cx x Dx
3 2 2 2 3 2 2 2
2 2 2 2
A B C D
A B C D
A B C D
A C
A = 0 , B = 1 / 4 , C = 0 , D =− 1 / 4
Assim,
C
x x
C
x x
dx x x
I
− −
1 1
2 2
( ) ( )
dx x x
x x
1 1
(^22)
2
( 1 ) ( 1 ) 1 ( 1 ) 1
2 2 2 2
2
x
Cx D
x
B
x
A
x x
x x
Calculando A,B,C e D
( ) ( ) ( ) ( ) ( )
2 2 2 2 x + 2 x − 1 ≡ Ax − 1 x + 1 + Bx + 1 + Cx + D x − 1
x = 1 → 2 = 2 B ∴ B = 1
x + 2 x − 1 ≡ A ( x + x − x − 1 ) + Bx + B + Cx − 2 Cx + Cx + Dx − 2 Dx + D
2 3 2 2 3 2 2
A B D
A C D
A B C D
A C
A = 1 , B = 1 , C =− 1 , D =− 1
1 2 3
1
x
y
dx x x
dx x x x x
A
3
2
3
2
Fazendo:
x
B
x
A
x x
Calculando A e B:
− 2 ≡ A ( x − 4 ) + B ( x − 1 )
= → − = ∴ =^ −
x A A
x B B
Assim,
x x C
dx x x x x
dx
ln 4 3
ln 1 3
ln 2.. 3
ln 2 3
ln 2 3
ln 1 ln 2 3
ln 2 ln 1 3
ln 4 3
ln 1 3
3
2
u a
A x x
- Calcular a área da região sob o gráfico de 2 5
2
x x
y de x =− 2 ate x = 2
A Figura que segue mostra a área.
-2 -1 1 2
x
y
2
2
2
2
2
2
2
2
arctg arctg u a
x arctg
x
dx
x x
dx
−
− −
- Calcular a área da região sob o gráfico
2 −
x x
y de x = 1 ate x = 4
A Figura que segue mostra a área.
- Calcular a área da região sob o gráfico de
( )
2 2 3
x
y de x =− 2 ate x = 2
A Figura que segue mostra a área.
-2 -1 1 2
-0.
x
y
( )
2 2 x 3
dx I
( )
2
2
2 2
2
3 sec
3 sec
tg
x tg
dx d
x tg
2 2
2
4 2
2
cos 9
9 sec
9 sec
3 sec
x x
x x arctg
sen C
d
d I d
θ θ
θ θ
θ
θ θ θ
θ
( ) (^ )
2
2
2
2
2 2 2
arctg u a
x
x x arctg x
dx A
− (^) −
- Investigar as integrais impróprias
(a)
+∞
(^10) x^2 x 5
dx I
( ) ( ) ( )
( ) ( ) ( )
2
2
2
2 2
(^12)
Ax x Bx C x
x x
Ax x Bx Cx
x
C
x
B
x
A
x x
x x
dx I
x = 5 ⇒ 1 = C. 25 ⇒ C =
x = 0 ⇒ 1 = B. − 5 ⇒ B =−
A
A
A
A
x A B C
