Question 1

Evaluate the integral.
-$$\int _ { \pi / 12 } ^ { \pi / 2 } ( 1 - \cos 4 x ) \cos 2 x d x$$

Accepted Answer

A)  $\frac { 13 } { 48 }$ 
B)  $\frac { 13 } { 24 }$ 
C)  $- \frac { 1 } { 24 }$ 
D)  $- \frac { 11 } { 24 }$ 
A)  $\frac { 13 } { 48 }$ 
B)  $\frac { 13 } { 24 }$ 
C)  $- \frac { 1 } { 24 }$ 
D)  $- \frac { 11 } { 24 }$

Question 2

Integrate the function.
-$$\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$$

Accepted Answer

A)  $\sqrt { 77 } - 77$ 
B)  $\frac { \sqrt { 77 } } { 77 }$ 
C)  $\frac { 2 \sqrt { 77 } } { 77 }$ 
D)  $77 ^ { 3 / 2 }$ 
A)  $\sqrt { 77 } - 77$ 
B)  $\frac { \sqrt { 77 } } { 77 }$ 
C)  $\frac { 2 \sqrt { 77 } } { 77 }$ 
D)  $77 ^ { 3 / 2 }$

Question 3

Evaluate the integral.
-$$\int - 8 x \cos 2 x d x$$

Accepted Answer

A)  $- 2 \cos 2 x - 4 \sin 2 x + C$ 
B)  $- 2 \cos 2 x - 4 x \sin 8 x + C$ 
C)  $- 2 \cos 2 x - 4 x \sin 2 x + C$ 
D)  $- 4 \cos 2 x - 8 x \sin 2 x + C$ 
A)  $- 2 \cos 2 x - 4 \sin 2 x + C$ 
B)  $- 2 \cos 2 x - 4 x \sin 8 x + C$ 
C)  $- 2 \cos 2 x - 4 x \sin 2 x + C$ 
D)  $- 4 \cos 2 x - 8 x \sin 2 x + C$

Question 4

Evaluate the integral.
-$$\int _ { 1 } ^ { 3 } \ln 4 x d x$$

Accepted Answer

A)  $4.07$ 
B)  $11.1$ 
C)  $8.07$ 
D)  $- 1.93$ 
A)  $4.07$ 
B)  $11.1$ 
C)  $8.07$ 
D)  $- 1.93$

Question 5

Evaluate the integral.
-$$\int x ^ { 3 } \cos 3 x d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } x ^ { 3 } \cos 3 x + \frac { 1 } { 3 } x ^ { 2 } \sin 3 x - \frac { 2 } { 9 } x \cos 3 x - \frac { 2 } { 27 } \sin 3 x + C$ 
B)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x - \frac { 1 } { 3 } x ^ { 2 } \cos 3 x + \frac { 2 } { 9 } x \sin 3 x + \frac { 2 } { 27 } \cos 3 x + C$ 
C)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x + 1 x ^ { 2 } \cos 3 x - 2 x \sin 3 x - 2 \cos 3 x + C$ 
D)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x + \frac { 1 } { 3 } x ^ { 2 } \cos 3 x - \frac { 2 } { 9 } x \sin 3 x - \frac { 2 } { 27 } \cos 3 x + C$ 
A)  $\frac { 1 } { 3 } x ^ { 3 } \cos 3 x + \frac { 1 } { 3 } x ^ { 2 } \sin 3 x - \frac { 2 } { 9 } x \cos 3 x - \frac { 2 } { 27 } \sin 3 x + C$ 
B)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x - \frac { 1 } { 3 } x ^ { 2 } \cos 3 x + \frac { 2 } { 9 } x \sin 3 x + \frac { 2 } { 27 } \cos 3 x + C$ 
C)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x + 1 x ^ { 2 } \cos 3 x - 2 x \sin 3 x - 2 \cos 3 x + C$ 
D)  $\frac { 1 } { 3 } x ^ { 3 } \sin 3 x + \frac { 1 } { 3 } x ^ { 2 } \cos 3 x - \frac { 2 } { 9 } x \sin 3 x - \frac { 2 } { 27 } \cos 3 x + C$

Question 6

Evaluate the integral.
-$$\int 15 x \cos \frac { 1 } { 2 } x d x$$

Accepted Answer

A)  $30 x \sin \left( \frac { 1 } { 2 } \right) x + 60 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
B)  $60 \sin \left( \frac { 1 } { 2 } \right) x - 30 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
C)  $15 x \sin \left( \frac { 1 } { 2 } \right) x - 30 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
D)  $15 \sin \left( \frac { 1 } { 2 } \right) x + 30 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
A)  $30 x \sin \left( \frac { 1 } { 2 } \right) x + 60 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
B)  $60 \sin \left( \frac { 1 } { 2 } \right) x - 30 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
C)  $15 x \sin \left( \frac { 1 } { 2 } \right) x - 30 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
D)  $15 \sin \left( \frac { 1 } { 2 } \right) x + 30 x \cos \left( \frac { 1 } { 2 } \right) x + C$

Question 7

Integrate the function.
-$$\int \frac { d x } { \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } }$$

Accepted Answer

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

Question 8

Evaluate the integral by using a substitution prior to integration by parts.
-$$\int x ^ { 2 } \sqrt { x + 5 } d x$$

Accepted Answer

A)  $\frac { \left( 15 x ^ { 2 } - 60 x + 200 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$ 
B)  $\frac { \left( 30 x ^ { 2 } - 120 x + 80 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$ 
C)  $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) \sqrt { ( x + 5 ) } } { 105 } + C$ 
D)  $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$ 
A)  $\frac { \left( 15 x ^ { 2 } - 60 x + 200 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$ 
B)  $\frac { \left( 30 x ^ { 2 } - 120 x + 80 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$ 
C)  $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) \sqrt { ( x + 5 ) } } { 105 } + C$ 
D)  $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$

Question 9

Evaluate the integral by using a substitution prior to integration by parts.
-$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 21 } } d x$

Accepted Answer

A)  $\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 21 } - \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
B)  $\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 21 } + \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
C)  $\frac { x } { 2 } \sqrt { x ^ { 2 } + 21 } + \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
D)  $\frac { x } { 2 } \sqrt { x ^ { 2 } + 21 } - \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
A)  $\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 21 } - \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
B)  $\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 21 } + \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
C)  $\frac { x } { 2 } \sqrt { x ^ { 2 } + 21 } + \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$ 
D)  $\frac { x } { 2 } \sqrt { x ^ { 2 } + 21 } - \frac { 21 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 21 } \right) + C$

Question 10

Use integration by parts to establish a reduction formula for the integral.
-$$\int \tan ^ { \mathrm { n } } \mathrm { xdx } , \mathrm { n } \neq 1$$

Accepted Answer

A)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 1 } x - \int \tan ^ { n - 2 } x d x$ 
B)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 1 } x + \int \tan ^ { n - 1 } x d x$ 
C)  $\int \tan ^ { n } x d x = \tan ^ { n - 1 } x - \frac { 1 } { n - 1 } \int \tan ^ { n - 2 } x d x$ 
D)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 2 } x - \int \tan ^ { n - 1 } x d x$ 
A)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 1 } x - \int \tan ^ { n - 2 } x d x$ 
B)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 1 } x + \int \tan ^ { n - 1 } x d x$ 
C)  $\int \tan ^ { n } x d x = \tan ^ { n - 1 } x - \frac { 1 } { n - 1 } \int \tan ^ { n - 2 } x d x$ 
D)  $\int \tan ^ { n } x d x = \frac { 1 } { n - 1 } \tan ^ { n - 2 } x - \int \tan ^ { n - 1 } x d x$

Question 11

Integrate the function.
-$$\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }$$

Accepted Answer

A)  $\frac { 2 } { x ^ { 2 } + 3 } + C$ 
B)  $9 \ln \left| \sqrt { 3 + x ^ { 2 } } \right| + C$ 
C)  $9 \ln \left| x + \sqrt { 3 + x ^ { 2 } } \right| + C$ 
D)  $x + \ln \left| 9 + \sqrt { 3 + x ^ { 2 } } \right| + C$ 
A)  $\frac { 2 } { x ^ { 2 } + 3 } + C$ 
B)  $9 \ln \left| \sqrt { 3 + x ^ { 2 } } \right| + C$ 
C)  $9 \ln \left| x + \sqrt { 3 + x ^ { 2 } } \right| + C$ 
D)  $x + \ln \left| 9 + \sqrt { 3 + x ^ { 2 } } \right| + C$

Question 12

Use various trigonometric identities to simplify the expression then integrate.
-$$\int \cos ^ { 6 } \theta \sin 2 \theta d \theta$$

Accepted Answer

A)  $- \frac { 1 } { 4 } \cos ^ { 8 } \theta + C$ 
B)  $\frac { 1 } { 8 } \cos ^ { 7 } \theta + C$ 
C)  $- \frac { 2 } { 9 } \cos ^ { 9 } \theta + C$ 
D)  $\frac { 2 } { 7 } \cos ^ { 6 } \theta + C$ 
A)  $- \frac { 1 } { 4 } \cos ^ { 8 } \theta + C$ 
B)  $\frac { 1 } { 8 } \cos ^ { 7 } \theta + C$ 
C)  $- \frac { 2 } { 9 } \cos ^ { 9 } \theta + C$ 
D)  $\frac { 2 } { 7 } \cos ^ { 6 } \theta + C$

Question 13

Use any method to evaluate the integral.
-$$\int \frac { 6 \csc ^ { 3 } x } { \tan x } d x$$

Accepted Answer

A)  $- 2 \csc ^ { 3 } x + C$ 
B)  $- 2 \csc ^ { 4 } x + C$ 
C)  $\frac { 3 } { 2 } \csc ^ { 4 } x \cot x + C$ 
D)  $- 2 \cot ^ { 3 } x + C$ 
A)  $- 2 \csc ^ { 3 } x + C$ 
B)  $- 2 \csc ^ { 4 } x + C$ 
C)  $\frac { 3 } { 2 } \csc ^ { 4 } x \cot x + C$ 
D)  $- 2 \cot ^ { 3 } x + C$

Question 14

Evaluate the integral.
-$$\int _ { - \pi / 8 } ^ { \pi / 8 } \tan ^ { 4 } 2 \mathrm { tdt }$$

Accepted Answer

A)  $\frac { \pi } { 6 } - \frac { 1 } { 3 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  $\frac { \pi } { 4 } - \frac { 2 } { 3 }$ 
A)  $\frac { \pi } { 6 } - \frac { 1 } { 3 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  $\frac { \pi } { 4 } - \frac { 2 } { 3 }$

Question 15

Evaluate the integral by using a substitution prior to integration by parts.
-$$\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x$$

Accepted Answer

A)  $x \ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) - 2 x + C$ 
B)  $\ln \left( 10 x + 10 x ^ { 2 } \right) - 2 x + C$ 
C)  $x \ln \left( 10 x + 10 x ^ { 2 } \right) + C$ 
D)  $\ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) + C$ 
A)  $x \ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) - 2 x + C$ 
B)  $\ln \left( 10 x + 10 x ^ { 2 } \right) - 2 x + C$ 
C)  $x \ln \left( 10 x + 10 x ^ { 2 } \right) + C$ 
D)  $\ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) + C$

Question 16

Evaluate the integral.
-$$\int \csc ^ { 3 } 6 t d t$$

Accepted Answer

A)  $- \frac { 1 } { 12 } \csc 6 t \cot 6 t - \frac { 1 } { 12 } \ln | \csc 6 t + \cot 6 t | + C$ 
B)  $\frac { 1 } { 12 } \csc 6 t \cot ^ { 2 } 6 t - \frac { 1 } { 12 } \ln | \csc 6 t + \cot 6 t | + C$ 
C)  $- \frac { 1 } { 2 } \csc 6 t \cot 6 t + \frac { 1 } { 2 } \ln | \csc 6 t + \cot 6 t | + C$ 
D)  $- \frac { 1 } { 2 } \csc 6 t \cot 6 t - \frac { 1 } { 2 } \ln | \csc 6 t + \cot 6 t | + \frac { t } { 6 } + C$ 
A)  $- \frac { 1 } { 12 } \csc 6 t \cot 6 t - \frac { 1 } { 12 } \ln | \csc 6 t + \cot 6 t | + C$ 
B)  $\frac { 1 } { 12 } \csc 6 t \cot ^ { 2 } 6 t - \frac { 1 } { 12 } \ln | \csc 6 t + \cot 6 t | + C$ 
C)  $- \frac { 1 } { 2 } \csc 6 t \cot 6 t + \frac { 1 } { 2 } \ln | \csc 6 t + \cot 6 t | + C$ 
D)  $- \frac { 1 } { 2 } \csc 6 t \cot 6 t - \frac { 1 } { 2 } \ln | \csc 6 t + \cot 6 t | + \frac { t } { 6 } + C$

Question 17

Use a trigonometric substitution to evaluate the integral.
-$$\int _ { 0 } ^ { \ln 3 } \frac { e ^ { t } d t } { 16 + e ^ { 2 t } }$$

Accepted Answer

A)  $- 0.100$ 
B)  $0.100$ 
C)  $0.067$ 
D)  $0.399$ 
A)  $- 0.100$ 
B)  $0.100$ 
C)  $0.067$ 
D)  $0.399$

Question 18

Evaluate the integral.
-Use the formula $\int \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \mathrm { dx } = \mathrm { xf } ^ { - 1 } ( \mathrm { x } ) - \int x \left( \frac { \mathrm { d } } { \mathrm { dx } } \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \right) \mathrm { dx }$ to evaluate the integral. $\int \sin ^ { - 1 } x d x$

Accepted Answer

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

Question 19

Evaluate the integral.
-$$\int _ { 0 } ^ { 3 } x ^ { 4 } \ln 6 x d x$$

Accepted Answer

A)  $- 45.71$ 
B)  $132.37$ 
C)  $150.19$ 
D)  $130.75$ 
A)  $- 45.71$ 
B)  $132.37$ 
C)  $150.19$ 
D)  $130.75$

Question 20

Integrate the function.
-$$\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } , x > 6$$

Accepted Answer

A)  $\frac { 1 } { 36 x } + C$ 
B)  $\ln \left| \frac { x } { 6 } + \frac { \sqrt { x ^ { 2 } - 36 } } { x } \right| + C$ 
C)  $\frac { 216 } { x } + C$ 
D)  $\ln \left| x + \sqrt { x ^ { 2 } - 36 } \right| + C$ 
A)  $\frac { 1 } { 36 x } + C$ 
B)  $\ln \left| \frac { x } { 6 } + \frac { \sqrt { x ^ { 2 } - 36 } } { x } \right| + C$ 
C)  $\frac { 216 } { x } + C$ 
D)  $\ln \left| x + \sqrt { x ^ { 2 } - 36 } \right| + C$

Evaluate the integral. - $\int _ { \pi / 12 } ^ { \pi / 2 } ( 1 - \cos 4 x ) \cos 2 x d x$

Integrate the function. - $\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$

Evaluate the integral. - $\int - 8 x \cos 2 x d x$

Evaluate the integral. - $\int _ { 1 } ^ { 3 } \ln 4 x d x$

Evaluate the integral. - $\int x ^ { 3 } \cos 3 x d x$

Evaluate the integral. - $\int 15 x \cos \frac { 1 } { 2 } x d x$

Integrate the function. - $\int \frac { d x } { \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } }$

Evaluate the integral by using a substitution prior to integration by parts. - $\int x ^ { 2 } \sqrt { x + 5 } d x$

Evaluate the integral by using a substitution prior to integration by parts. - $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 21 } } d x$

Use integration by parts to establish a reduction formula for the integral. - $\int \tan ^ { \mathrm { n } } \mathrm { xdx } , \mathrm { n } \neq 1$

Integrate the function. - $\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }$

Use various trigonometric identities to simplify the expression then integrate. - $\int \cos ^ { 6 } \theta \sin 2 \theta d \theta$

Use any method to evaluate the integral. - $\int \frac { 6 \csc ^ { 3 } x } { \tan x } d x$

Evaluate the integral. - $\int _ { - \pi / 8 } ^ { \pi / 8 } \tan ^ { 4 } 2 \mathrm { tdt }$

Evaluate the integral by using a substitution prior to integration by parts. - $\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x$

Evaluate the integral. - $\int \csc ^ { 3 } 6 t d t$

Use a trigonometric substitution to evaluate the integral. - $\int _ { 0 } ^ { \ln 3 } \frac { e ^ { t } d t } { 16 + e ^ { 2 t } }$

Evaluate the integral. - $\int _ { 0 } ^ { 3 } x ^ { 4 } \ln 6 x d x$

Integrate the function. - $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } , x > 6$

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Exam 8: Techniques of Integration

Evaluate the integral. - ∫π/12π/2(1−cos⁡4x)cos⁡2xdx\int _ { \pi / 12 } ^ { \pi / 2 } ( 1 - \cos 4 x ) \cos 2 x d x∫π/12π/2​(1−cos4x)cos2xdx

Integrate the function. - ∫0281dx(81−x2)3/2\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }∫02​(81−x2)3/281dx​

Evaluate the integral. - ∫−8xcos⁡2xdx\int - 8 x \cos 2 x d x∫−8xcos2xdx

Evaluate the integral. - ∫13ln⁡4xdx\int _ { 1 } ^ { 3 } \ln 4 x d x∫13​ln4xdx

Evaluate the integral. - ∫x3cos⁡3xdx\int x ^ { 3 } \cos 3 x d x∫x3cos3xdx

Evaluate the integral. - ∫15xcos⁡12xdx\int 15 x \cos \frac { 1 } { 2 } x d x∫15xcos21​xdx

Integrate the function. - ∫dx(x2+4)3/2\int \frac { d x } { \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } }∫(x2+4)3/2dx​

Evaluate the integral by using a substitution prior to integration by parts. - ∫x2x+5dx\int x ^ { 2 } \sqrt { x + 5 } d x∫x2x+5​dx

Evaluate the integral by using a substitution prior to integration by parts. - ∫x2x2+21dx\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 21 } } d x∫x2+21​x2​dx

Use integration by parts to establish a reduction formula for the integral. - ∫tan⁡nxdx,n≠1\int \tan ^ { \mathrm { n } } \mathrm { xdx } , \mathrm { n } \neq 1∫tannxdx,n=1

Integrate the function. - ∫9dx3+x2\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }∫3+x2​9dx​

Use various trigonometric identities to simplify the expression then integrate. - ∫cos⁡6θsin⁡2θdθ\int \cos ^ { 6 } \theta \sin 2 \theta d \theta∫cos6θsin2θdθ

Use any method to evaluate the integral. - ∫6csc⁡3xtan⁡xdx\int \frac { 6 \csc ^ { 3 } x } { \tan x } d x∫tanx6csc3x​dx

Evaluate the integral. - ∫−π/8π/8tan⁡42tdt\int _ { - \pi / 8 } ^ { \pi / 8 } \tan ^ { 4 } 2 \mathrm { tdt }∫−π/8π/8​tan42tdt

Evaluate the integral by using a substitution prior to integration by parts. - ∫ln⁡(10x+10x2)dx\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x∫ln(10x+10x2)dx

Evaluate the integral. - ∫csc⁡36tdt\int \csc ^ { 3 } 6 t d t∫csc36tdt

Use a trigonometric substitution to evaluate the integral. - ∫0ln⁡3etdt16+e2t\int _ { 0 } ^ { \ln 3 } \frac { e ^ { t } d t } { 16 + e ^ { 2 t } }∫0ln3​16+e2tetdt​

Evaluate the integral. - ∫03x4ln⁡6xdx\int _ { 0 } ^ { 3 } x ^ { 4 } \ln 6 x d x∫03​x4ln6xdx

Integrate the function. - ∫dxx2x2−36,x>6\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } , x > 6∫x2x2−36​dx​,x>6