Deck 5: Techniques of Integration

Question 1

Find $\int x \left( x ^ { 2 } - 1 \right) ^ { 4 } d x$

A) $\frac { 1 } { 8 } \left( x ^ { 2 } - 1 \right) ^ { 4 } + C$
B) $\frac { 1 } { 8 } \left( x ^ { 2 } + 1 \right) ^ { 4 } + C$
C) $\frac { 1 } { 10 } \left( x ^ { 2 } - 1 \right) ^ { 5 } + C$
D) $\frac { 1 } { 10 } \left( x ^ { 2 } + 1 \right) ^ { 5 } + C$

Answer 1

$\frac { 1 } { 10 } \left( x ^ { 2 } - 1 \right) ^ { 5 } + C$

Question 2

Find $\int \frac { 5 x } { 2 x ^ { 2 } + 1 } d x$

A) $\frac { 5 } { 2 } \ln \left( 2 x ^ { 2 } + 1 \right) + C$

B) $\frac { 5 } { 4 } \ln \left( 2 x ^ { 2 } + 1 \right) + C$

C) $\frac { 5 } { 2 } \ln \left( 2 x ^ { 2 } - 1 \right) + C$

D) $\frac { 5 } { 4 } \ln \left( 2 x ^ { 2 } - 1 \right) + C$

Answer 2

$\frac { 5 } { 4 } \ln \left( 2 x ^ { 2 } + 1 \right) + C$

Question 3

Find $\int \frac { \sqrt { x } - 5 } { \sqrt { x } } d x$

A) $x - \frac { 10 } { \sqrt { x } } + C$
B) $x - 5 \sqrt { x } + C$
C) $x + 10 \sqrt { x } + C$
D) $x - 10 \sqrt { x } + C$

Answer 3

$x - 10 \sqrt { x } + C$

Question 4

Find $\int x ^ { 4 } \sin \left( x ^ { 5 } \right) d x$

A) $\frac { 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$
B) $\frac { 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$
C) $\frac { - 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$
D) $\frac { - 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$

Question 5

Find $\int \frac { \sin ( \ln x ) } { x } d x$

A) $\cos ( \ln x ) + C$
B) $- \cos ( \ln x ) + C$
C) $\sin ( \ln x ) + C$
D) $- \sin ( \ln x ) + C$

Question 6

Find $\int x ^ { 2 } \ln \left( e ^ { x ^ { 3 } + 2 } \right) d x$

A) $\frac { 1 } { 6 } \left( x ^ { 3 } - 2 \right) ^ { 2 } + C$
B) $\frac { 1 } { 6 } \left( x ^ { 2 } + 2 \right) ^ { 2 } + C$
C) $\frac { 1 } { 6 } \left( x ^ { 3 } + 2 \right) ^ { 3 } + C$
D) $\frac { 1 } { 6 } \left( x ^ { 3 } + 2 \right) ^ { 2 } + C$

Question 7

Find $\int \cos ( x ) e ^ { - 2 \sin ( x ) } d x$

A) $\frac { 1 } { 2 } e ^ { - 2 \sin x } + C$
B) $\frac { - 1 } { 2 } e ^ { - 2 \cos x } + C$
C) $- e ^ { - 2 \sin x } + C$
D) $\frac { - 1 } { 2 } e ^ { - 2 \sin x } + C$

Question 8

Find $\int \sec ( 3 x ) \tan ( 3 x ) d x$

A) $\sec 3 x + C$
B) $\frac { 1 } { 2 } \sec ^ { 2 } 3 x + C$
C) $\frac { 1 } { 3 } \sec 3 x \tan 3 x + C$
D) $\frac { 1 } { 3 } \sec 3 x + C$

Question 9

Find $\int \frac { \sqrt [ 3 ] { \ln x } } { x } d x$

A) $\frac { 1 } { 4 } ( \ln x ) ^ { 1 / 3 } + C$
B) $\frac { 3 } { 4 } ( \ln x ) ^ { 4 / 3 } + C$
C) $( \ln x ) ^ { 1 / 3 } + C$
D) $\ln x + C$

Question 10

Find $\int \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$

A) $\frac { 1 } { ( \ln x ) ^ { 2 } } + C$
B) $\frac { 1 } { 3 ( \ln x ) ^ { 3 } } + C$
C) $\frac { - 1 } { 2 ( \ln x ) ^ { 2 } } + C$
D) $\frac { 1 } { 2 ( \ln x ) ^ { 2 } } + C$

Question 11

Find $\int \frac { \csc ^ { 2 } ( x ) } { 1 + 2 \cot ( x ) } d x$

A) $\ln | 1 + 2 \cot x | + C$
B) $\frac { 1 } { 2 } \ln | 1 + 2 \cot x | + C$
C) $\frac { - 1 } { 2 } \ln | 1 + 2 \cot x | + C$
D) $- \ln | 1 + 2 \cot x | + C$

Question 12

Find $\int \frac { \sin ( \sqrt { x } ) } { \sqrt { x } } d x$

A) $- 2 \sin \sqrt { x } + C$

B) $2 \cos \sqrt { x } + C$

C) $2 \sin \sqrt { x } + C$

D) $- 2 \cos \sqrt { x } + C$

Question 13

Find $\int \tan ( x ) \ln \left( e ^ { \cos ( x ) } \right) d x$

A) $\ln | \cos x | + C$
B) $- \ln | \cos x | + C$
C) sin x + C
D) -cos x + C

Question 14

Find $\int \frac { e ^ { x } } { e ^ { 2 x } + 1 } d x$

A) $\frac { 1 } { \left( e ^ { x } + 1 \right) ^ { 2 } } + C$
B) $\frac { 1 } { \left( e ^ { x } + 1 \right) ^ { 2 } } + C$
C) $\arctan \left( e ^ { x } \right) + C$
D) $\arctan \left( e ^ { 2 x } \right) + C$

Question 15

Find $\int \frac { e ^ { 3 x } } { e ^ { 3 x } + 4 } d x$

A) $\ln \left( e ^ { 3 x } + 4 \right) + C$
B) $\frac { 1 } { 3 } \ln \left( e ^ { 3 x } - 4 \right) + C$
C) $\frac { 1 } { 3 \left( e ^ { 3 x } + 4 \right) ^ { 2 } } + C$
D) $\frac { 1 } { 3 } \ln \left( e ^ { 3 x } + 4 \right) + C$

Question 16

Find $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x$

A) $\frac { 2 } { 3 } \left( x ^ { 3 } + 2 \right) ^ { 1 / 2 } + C$
B) $\frac { 2 } { 9 } \left( x ^ { 3 } + 2 \right) ^ { 3 / 2 } + C$
C) $\frac { 2 } { 3 } \left( x ^ { 3 } + 2 \right) ^ { 3 / 2 } + C$
D) $\frac { 2 } { 9 } \left( x ^ { 3 } + 2 \right) ^ { 1 / 2 } + C$

Question 17

Find $\int \frac { x } { \sqrt { x - 2 } } d x$

Question 18

Find $\int x \sqrt { x + 3 } d x$

Question 19

Find $\int \frac { e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } } d x$

A) $\sin ^ { - 1 } \left( e ^ { 2 x } \right) + C$
B) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( e ^ { 2 x } \right) + C$
C) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( e ^ { x } \right) + C$
D) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( e ^ { 2 x } \right) + C$

Question 20

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 2 \right) \sqrt { x } d x$

A) 12/7
B) 14/5
C) 14/9
D) 12/9

Question 21

Evaluate $\int _ { 0 } ^ { 2 } x \sqrt { 2 - x } d x$

A) $\frac { 16 } { 15 } \sqrt { 3 }$
B) $\frac { 16 } { 13 } \sqrt { 2 }$
C) $\frac { 15 } { 16 } \sqrt { 2 }$
D) $\frac { 16 } { 15 } \sqrt { 2 }$

Question 22

Evaluate $\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 2 + e ^ { x } } d x$

Question 23

Evaluate $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x$

A) $\pi$ /2
B) $\pi$ /4
C) 3 $\pi$ /4
D) $\pi$

Question 24

Evaluate $\int _ { 0 } ^ { \sqrt { \pi / 2 } } x \cos \left( x ^ { 2 } \right) d x$

A) $\frac { \sqrt { 3 } } { 2 }$

B) - $\frac { \sqrt { 3 } } { 2 }$

C) $1 / 2$

D) - $1 / 2$

Question 25

Evaluate $\int _ { 0 } ^ { \pi / 4 } \tan ( x ) \sec ^ { 2 } ( x ) d x$

A) $\sqrt { 3 }$
B) $\frac { \sqrt { 2 } } { 2 }$
C) 1
D) $1 / 2$

Question 26

Find $\int x ^ { 2 } e ^ { x } d x$

A) $x ^ { 2 } e ^ { x } + 2 x e ^ { x } + 2 e ^ { x } + C$

B) $x ^ { 2 } e ^ { x } - 2 x e ^ { x } - 2 e ^ { x } + C$

C) $x ^ { 2 } e ^ { x } - 2 x e ^ { x } + 2 e ^ { x } + C$

D) $x ^ { 2 } e ^ { x } + x e ^ { x } - 2 e ^ { x } + C$

Question 27

Find $\int x e ^ { 4 x } d x$

A) $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 16 } e ^ { 4 x } + C$
B) $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 4 } e ^ { 4 x } + C$
C) $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 4 } e ^ { 4 x } + C$
D) $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 16 } e ^ { 4 x } + C$

Question 28

Find $\int \frac { x ^ { 2 } } { e ^ { 2 x } } d x$

A) $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } + \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$
B) $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } + \frac { 1 } { 4 } e ^ { - 2 x } + C$
C) $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$
D) $\frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$

Question 29

Find $\int x ^ { 2 } \sin ( x ) d x$

A) $x ^ { 2 } \cos x - 2 x \sin x + 2 \cos x + C$
B) $x ^ { 2 } \cos x + 2 x \sin x + 2 \cos x + C$
C) $- x ^ { 2 } \cos x - 2 x \sin x + 2 \cos x + C$
D) $- x ^ { 2 } \cos x + 2 x \sin x + 2 \cos x + C$

Question 30

Find $\int x \cos ( 3 x ) d x$

A) $\frac { 1 } { 3 } x \sin 3 x - \frac { 1 } { 9 } \cos 3 x + C$
B) $\frac { 1 } { 3 } x \cos 3 x + \frac { 1 } { 9 } \sin 3 x + C$
C) $- \frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$
D) $\frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$

Question 31

Find $\int x ^ { 2 } \ln x d x$

A) $\frac { 1 } { 3 } x ^ { 3 } \ln x + \frac { 1 } { 9 } x ^ { 3 } + C$
B) $\frac { 1 } { 3 } x ^ { 3 } \ln x - \frac { 1 } { 3 } x ^ { 3 } + C$
C) $\frac { 1 } { 3 } x ^ { 2 } \ln x + \frac { 1 } { 3 } x ^ { 3 } + C$
D) $\frac { 1 } { 3 } x ^ { 3 } \ln x - \frac { 1 } { 9 } x ^ { 3 } + C$

Question 32

Find $\int x \sec ^ { 2 } ( x ) d x$

A) $x \sec x \tan x + \ln | \cos x | + C$
B) $x \tan x + \ln | \cos x | + C$
C) $x \tan x - \ln | \cos x | + C$
D) $x \tan x + \ln | \sin x | + C$

Question 33

Find $\int \ln \left( x ^ { 2 } \right) d x$

A) $2 x ( \ln x + 1 ) + C$
B) $2 x \ln x - x + C$
C) $2 x ( \ln x - 1 ) + C$
D) $2 x \ln x + x + C$

Question 34

Find $\int \ln ( 5 x ) d x$

A) $x \ln 5 x - x + C$
B) $x \ln 5 x + x + C$
C) $x \ln x - x + C$
D) $5 x \ln x - x + C$

Question 35

Find $\int \tan ^ { - 1 } ( 2 x ) d x$

A) $x \tan ^ { - 1 } ( 2 x ) - \ln \left( 1 + 4 x ^ { 2 } \right) + C$
B) $x \tan ^ { - 1 } ( 2 x ) - \frac { 1 } { 4 } \ln \left( 1 + 2 x ^ { 2 } \right) + C$
C) $x \tan ^ { - 1 } ( 2 x ) - \ln \left( 1 + 2 x ^ { 2 } \right) + C$
D) $x \tan ^ { - 1 } ( 2 x ) - \frac { 1 } { 4 } \ln \left( 1 + 4 x ^ { 2 } \right) + C$

Question 36

Find $\int \sin ^ { - 1 } ( 3 x ) d x$

Question 37

Find $\int e ^ { x } \ln \left( e ^ { 3 x } \right) d x$

Question 38

Find $\int x \ln 3 x d x$

Question 39

Find $\int x ^ { 3 } \cos \left( x ^ { 2 } \right) d x$

A) $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$
B) $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$
C) $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$
D) $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$

Question 40

Find $\int e ^ { x } \sin ( 2 x ) d x$

A) $\frac { 2 } { 5 } e ^ { x } \cos 2 x + \frac { 1 } { 5 } e ^ { x } \sin 2 x + C$

B) $- \frac { 2 } { 5 } e ^ { x } \cos 2 x - \frac { 1 } { 5 } e ^ { x } \sin 2 x + C$

C) $- \frac { 2 } { 5 } e ^ { x } \cos 2 x + \frac { 1 } { 5 } e ^ { x } \sin 2 x + C$

D) $- \frac { 1 } { 5 } e ^ { x } \cos 2 x + \frac { 1 } { 5 } e ^ { x } \sin 2 x + C$

Question 41

Find $\int e ^ { - x } \cos ( x ) d x$

Question 42

Evaluate $\int _ { - 1 } ^ { 1 } \tan ^ { - 1 } ( x ) d x$

A) 3ln3
B) 3ln3 + 1
C) 0
D) $\frac { \pi } { 2 }$

Question 43

Evaluate $\int _ { 0 } ^ { \pi 2 } e ^ { x } \cos ( x ) d x$

A) 1
B) 0
C) $\frac { 1 } { 2 } e ^ { \pi / 2 }$
D) $\frac { 1 } { 2 } e ^ { \pi / 2 } - \frac { 1 } { 2 }$

Question 44

Evaluate $\int _ { 1 }^{3} \ln x d x$

Question 45

Find $\int \frac { 4 } { ( x + 2 ) ( x - 1 ) } d x$

Question 46

Find $\int \frac { 3 x + 2 } { x ( x - 2 ) } d x$

Question 47

Find $\int \frac { x - 2 } { ( x + 1 ) ^ { 2 } } d x$

Question 48

Find $\int \frac { 2 x + 1 } { x ^ { 2 } ( x + 2 ) } d x$

Question 49

Find $\int \frac { 5 } { x ^ { 2 } - 1 } d x$

Question 50

Find $\int \frac { 1 - x } { x ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x$

Question 51

Find $\int \frac { x + 4 } { x ^ { 2 } - 3 x - 4 } d x$

Question 52

Find $\int \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 3 } + 2 x ^ { 2 } } d x$

Question 53

Find $\int \frac { x } { \left( x ^ { 2 } + 4 \right) ( x - 1 ) } d x$

Question 54

Evaluate $\int _ { - 1 } ^ { 2 } \frac { 2 } { x ^ { 2 } - 9 } d x$

Question 55

Evaluate $\int _ { 2 } ^ { 3 } \frac { 3 } { x ( x - 1 ) } d x$

A) $4 \ln \left( \frac { 4 } { 3 } \right)$
B) $3 \ln \left( \frac { 3 } { 4 } \right)$
C) $3 \ln \left( \frac { 4 } { 3 } \right)$
D) $4 \ln \left( \frac { 3 } { 4 } \right)$

Question 56

Evaluate $\int _ { 2 } ^ { 3 } \frac { 2 \left( x ^ { 2 } - 4 \right) } { x ^ { 2 } - 1 } d x$

A) $3 + 3 \ln \left( \frac { 2 } { 3 } \right)$
B) $3 \ln \left( \frac { 2 } { 3 } \right)$
C) $3 \ln \left( \frac { 4 } { 3 } \right)$
D) $2 + 3 \ln \left( \frac { 2 } { 3 } \right)$

Question 57

Find the integral using substitution and integration by parts $\int \frac { \cos ( x ) } { \sin ( x ) + \sin ^ { 2 } ( x ) } d x$

Question 58

Find $\int \sin ^ { 2 } ( x ) d x$

Question 59

Find $\int \cos ^ { 2 } ( 2 x ) d x$

Question 60

Find $\int \cos ^ { 3 } ( x ) d x$

Question 61

Find $\int \sin ^ { 5 } ( x ) d x$

Question 62

Find $\int \sec ^ { 4 } ( x ) d x$

Question 63

Find $\int \sin ^ { 3 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$

Question 64

Find $\int \cos ^ { 5 } ( x ) \sin ^ { 4 } ( x ) d x$

Question 65

Find $\int \sec ^ { 5 } ( x ) \tan ( x ) d x$

A) $\frac { 1 } { 5 } \tan ^ { 5 } x + C$
B) $\frac { 1 } { 5 } \sec ^ { 5 } x + C$
C) $\frac { 1 } { 5 } \sec ^ { 5 } x \tan ^ { 5 } x + C$
D) $\frac { 1 } { 3 } \sec ^ { 3 } x + C$

Question 66

Find $\int \sec ^ { 6 } ( x ) \tan ( x ) d x$

Question 67

Find $\int \sec ( x ) \tan ^ { 5 } ( x ) d x$

Question 68

Find $\int \sin ^ { 3 } ( x ) \csc ( x ) d x$

Question 69

Find $\int \sec ( x ) \cos ^ { 4 } ( x ) d x$

Question 70

Find $\int \sin ^ { 5 } ( x ) \cot ( x ) d x$

A) $\frac { 1 } { 5 } \sec ^ { 5 } x + C$
B) $\frac { 1 } { 5 } \cos ^ { 5 } x + C$
C) $\frac { 1 } { 5 } \sin ^ { 5 } x + C$
D) $\frac { 1 } { 6 } \sin ^ { 6 } x + C$

Question 71

Find $\int \cos ^ { 3 } ( x ) \tan ( x ) d x$

A) $\frac { 1 } { 5 } \sin ^ { 5 } x + C$
B) $\frac { 1 } { 3 } \sin ^ { 3 } x + C$
C) $\frac { 1 } { 3 } \cos ^ { 3 } x + C$
D) $- \frac { 1 } { 3 } \cos ^ { 3 } x + C$

Question 72

Evaluate $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( x ) d x$

Question 73

Evaluate $\int _ { - \pi } ^ { \pi } \sin ^ { 3 } ( x ) d x$

A) 1
B) 0
C) $1 / 2$
D) -1

Question 74

Evaluate $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 3 } ( x ) \sin ^ { 2 } ( x ) d x$

A) 2/15
B) 1/5
C) 4/15
D) 1/3

Question 75

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \sin ( x ) } { \cos ^ { 2 } ( x ) } d x$

A) -1
B) 0
C) 1
D) $1 / 2$

Question 76

Evaluate $\int _ { - \pi / 4 } ^ { 0 } \frac { \sin ^ { 2 } ( x ) } { \cos ^ { 4 } ( x ) } d x$

Question 77

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \tan ( x ) } { \cos ^ { 3 } ( x ) } d x$

Question 78

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $\tan \left( \sec ^ { - 1 } ( x ) \right)$

A) $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$
B) $\sqrt { 1 - x ^ { 2 } }$
C) $\sqrt { x ^ { 2 } - 1 }$
D) $\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }$

Question 79

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $$\sin \left( \cos ^ { - 1 } ( x ) \right)$$

Question 80

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $$\tan \left( \sin ^ { - 1 } ( x ) \right)$$