Deck 9: Techniques of Integration

Question 1

Determine the integral by making an appropriate substitution.
dx

A) + C
B) - + C
C) -x + C
D) x + C
E) none of these

Answer 1

- + C

Question 2

Determine the integral by making an appropriate substitution.
dx Use the substitution u = ln 5x.

A) + C
B) (ln 5x + C
C) + C
D) (ln x) + C
E) none of these

Answer 2

(ln 5x + C

Question 3

Determine the integral by making an appropriate substitution.

- $\int 4 ( 2 x + 5 ) ^ { 3 } d x$

A) $\frac { 1 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C
B) $\frac { 3 } { 8 }$ $( 2 x + 5 ) ^ { 4 }$ + C
C) $\frac { 3 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C
D) $\frac { 1 } { 2 }$ $( 2 x + 5 ) ^ { 4 }$ + C

Answer 3

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

Question 4

Determine the integral by making an appropriate substitution.

A) ln + C
B) sin x cos x + C
C) -ln + C
D) + C
E) none of these

Question 5

Determine the integral by making an appropriate substitution.
dx

A)
B) + C
C)
D) + C

Question 6

Determine the integral by making an appropriate substitution.
dx

A) (x + 2) + C
B) + C
C) 6 + C
D) (x - 2) + C

Question 7

Determine the integral by making an appropriate substitution.

A) -9 + C
B) + C
C) 9 + C
D) - + C

Question 8

Determine the integral by making an appropriate substitution.
dx

A) - + C
B) - + C
C)
D) - + C

Question 9

Determine the integral by making an appropriate substitution.
dx Use the substitution u =ln .

A) + C
B) 2(ln + C
C) + C
D) + C
E) none of these

Question 10

Determine the integral by making an appropriate substitution.
dx

A) + C
B) + C
C) + C
D) + C

Question 11

Determine the integral by making an appropriate substitution.
dx

A) x sin x + C
B) x + C
C) - x + C
D) x + C
E) none of these

Question 12

Determine the integral by making an appropriate substitution.

- $\int \frac { \sec ^ { 2 } x } { ( 1 - 3 \tan x ) ^ { 1 / 2 } }$ dx

A) ln $| 1 - 3 \tan x |$ + C
B) - $\frac { 2 } { 3 }$ ln $| 1 - 3 \tan x |$ + C
C) - $\frac { 2 } { 3 }$ $( 1 - 3 \tan x ) ^ { 1 / 2 }$ + C
D) $( 1 - 3 \tan x ) ^ { 1 / 2 }$ + C
E) none of these

Question 13

Determine the integral by making an appropriate substitution.
dx

A) + C
B) + C
C) + C
D) + C

Question 14

Determine the integral by making an appropriate substitution.

A) + C
B) + C
C) + C
D) 1/2 + C

Question 15

Determine the integral by making an appropriate substitution.

A) + + C
B) + + C
C) + C
D) + C

Question 16

Determine the integral by making an appropriate substitution.
dθ

A) + C
B) - ln + C
C) + C
D) θ θ + C
E) none of these

Question 17

Determine the integral by making an appropriate substitution.
dx

A) + C
B) + C
C) ln + C
D) ln + C
E) none of these

Question 18

Determine the integral by making an appropriate substitution.

- $\int 12 \sin x \sqrt { \cos x }$ dx

A) 18 $( \cos x ) ^ { 3 / 2 }$ + C
B) 8 $( \cos x ) ^ { 3 / 2 }$ + C
C) -18 $( \cos x ) ^ { 3 / 2 }$ + C
D) -8 $( \cos x ) ^ { 3 / 2 }$ + C

Question 19

Determine the integral by making an appropriate substitution.
Use the substitution u = tan x.

A) + C
B) sec x tan x + C
C) + C
D) + C
E) none of these

Question 20

Determine the integral by making an appropriate substitution.

A) 21 cos(7 - 6) + C
B) - cos(7 - 6) + C
C) cos(7 - 6) + C
D) -21 cos(7 - 6) + C

Question 21

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 22

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 23

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 24

Determine the integral by making an appropriate substitution.
dx

A) tan + C
B) + C
C) sec + C
D) tan + C
E) none of these

Question 25

Determine the integral by making an appropriate substitution.
dx

A) + C
B) + C
C) + C
D) + C
E) none of these

Question 26

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 27

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 28

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 29

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 30

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 31

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 32

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 33

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 34

Determine the integral by making an appropriate substitution.

A) + C
B) + C
C) ln + C
D) ln + C
E) none of these

Question 35

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 36

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 37

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 38

Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Question 39

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 40

Determine the integral by making an appropriate substitution.
dx

A) x + C
B) x + C
C) 9 x + C
D) x + C
E) none of these

Question 41

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 42

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form . No parentheses around arguments of functions.

Question 43

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 44

Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 45

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 46

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 47

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 48

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 49

Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 50

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 51

Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 52

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 53

dx [Hint: Let u = ln(ln x).]
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 54

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 55

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 56

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 57

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 58

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 59

Enter your answer with any coefficients in front as integers or reduced fractions of form . No parentheses around arguments of functions.

Question 60

Enter your answer with any coefficients in front as integers or reduced fractions of form .

Question 61

Evaluate the integral using integration by parts.

- $\int x$ $\sqrt { x + 1 }$ dx

A) $\frac { 2 } { 3 }$ $( x + 1 ) ^ { 3 / 2 }$ - $\frac { 4 } { 15 }$ $( x + 1 ) ^ { 5 / 2 }$ + C
B) $\frac { 2 } { 3 }$ x $( x + 1 ) ^ { 3 / 2 }$ - $\frac { 4 } { 15 }$ $( x + 1 ) ^ { 5 / 2 }$ + C
C) x $( x + 1 ) ^ { 3 / 2 }$ - $( x + 1 ) ^ { 5 / 2 }$ + C
D) $( x + 1 ) ^ { 3 / 2 }$ + $( x + 1 ) ^ { 5 / 2 }$ + C

Question 62

Evaluate the integral using integration by parts.

A) + C
B) + C
C) + C
D) + C

Question 63

Evaluate the integral using integration by parts.

A) + C
B) + C
C) + C
D) + C

Question 64

Evaluate the integral using integration by parts.

A) bx + C
B) (sin bx - cos bx) + C
C) + C
D) (sin bx + cos bx) + C

Question 65

Evaluate the integral using integration by parts.
dx

A) x + + C
B) x + + C
C) x x + + C
D) x - + C

Question 66

Evaluate the integral using integration by parts.
dx

A) + 1 + C
B) + C
C) + C
D) + C

Question 67

Evaluate the integral using integration by parts.

A) - x - + C
B) - x + + C
C) x + + C
D) x - + C

Question 68

Which of the following is a correct substitution for the integral ?

A) u = ln x;
B) u = ;
C) u = ln x;
D) u= ln x; du
E) u = ln x;

Question 69

Does this integral = 2 (x - 1) + C?

Question 70

Evaluate the integral using integration by parts.

A) (sin 3x - cos 3x) + C
B) - (sin 3x - 3 cos 3x) + C
C) (sin 3x - 3 cos 3x) + C
D) (2 sin 3x - 3 cos 3x) + C

Question 71

Evaluate the integral using integration by parts.

A) + C
B) + C
C) + C
D) + C
E) none of these

Question 72

Evaluate the integral using integration by parts.

A) x tan x + tan x + C
B) 4 x tan x+ C
C) x tan x + C
D) x tan x + tan x + C

Question 73

x dx [Hint: x = 1 - x.]
Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 74

Evaluate the integral using integration by parts.

A) (sin bx - cos bx) + C
B) (sin bx + cos bx) + C
C) bx + C
D) + C

Question 75

dx
Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 76

Evaluate the integral using integration by parts.

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

A) -(2x +2) $\mathrm { e } ^ { - \mathrm { x } }$ + C
B) ( $x ^ { 2 }$ + 2x +2) $\mathrm { e } ^ { - \mathrm { x } }$ + C
C) ( $x ^ { 2 }$ + 2x) $\mathrm { e } ^ { - \mathrm { x } }$ + C
D) -( $x ^ { 2 }$ + 2x +2) $\mathrm { e } ^ { - \mathrm { x } }$ + C

Question 77

Evaluate the integral using integration by parts.
dx

A) + C
B) (8x - 1) + C
C) + C
D) x + C

Question 78

Evaluate the integral using integration by parts.

A) x + C
B) (x + 2) + C
C) (x + 1) + C
D) + C

Question 79

dx [Hint: csc x = -cot x csc x]
Enter your answer with any coefficients in front as integers or reduced fractions of form .
No parentheses around arguments of functions.

Question 80

Evaluate the integral using integration by parts.

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