Question 1

Integrate the function.
-$$\int _ { - 1 } ^ { 1 } \frac { 6 } { 1 + 36 t ^ { 2 } } d t$$

Accepted Answer

A)  $2 \tan ^ { - 1 } 6$ 
B)  $\frac { \pi } { 2 }$ 
C)  $2 \tan ^ { - 1 } \left( \frac { 1 } { 6 } \right)$ 
D)  $2 \sin ^ { - 1 } 6$ 
A)  $2 \tan ^ { - 1 } 6$ 
B)  $\frac { \pi } { 2 }$ 
C)  $2 \tan ^ { - 1 } \left( \frac { 1 } { 6 } \right)$ 
D)  $2 \sin ^ { - 1 } 6$

Question 2

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos 2 x } d x$$

Accepted Answer

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

Question 3

Use integration by parts to establish a reduction formula for the integral.
-$$\int x ^ { n } e ^ { - a x } d x$$

Accepted Answer

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

Question 4

Use various trigonometric identities to simplify the expression then integrate.
-$\int \sin ^ { 2 } \theta \cos 5 \theta d \theta$

Accepted Answer

A)  $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$ 
B)  $\frac { 1 } { 2 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 7 } \sin 7 \theta + C$ 
C)  $\frac { 1 } { 10 } \sin 3 \theta - \frac { 1 } { 4 } \sin 5 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$ 
D)  $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 7 \theta - \frac { 1 } { 28 } \sin 3 \theta + C$ 
A)  $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$ 
B)  $\frac { 1 } { 2 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 7 } \sin 7 \theta + C$ 
C)  $\frac { 1 } { 10 } \sin 3 \theta - \frac { 1 } { 4 } \sin 5 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$ 
D)  $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 7 \theta - \frac { 1 } { 28 } \sin 3 \theta + C$

Question 5

Use any method to evaluate the integral.
-$$\int 6 x \sin ^ { 2 } x d x$$

Accepted Answer

A)  $3 x ^ { 2 } - \frac { 3 } { 2 } \sin 2 x - \frac { 3 } { 4 } \cos 2 x + C$ 
B)  $\frac { 3 } { 2 } x ^ { 2 } - \frac { 3 } { 2 } \sin 2 x - \frac { 3 } { 4 } \cos 2 x + C$ 
C)  $6 x ^ { 2 } - 3 \sin 2 x - \frac { 3 } { 2 } \cos 2 x + C$ 
D)  $3 x ^ { 2 } + \frac { \sin ^ { 3 } x } { 3 } + C$ 
A)  $3 x ^ { 2 } - \frac { 3 } { 2 } \sin 2 x - \frac { 3 } { 4 } \cos 2 x + C$ 
B)  $\frac { 3 } { 2 } x ^ { 2 } - \frac { 3 } { 2 } \sin 2 x - \frac { 3 } { 4 } \cos 2 x + C$ 
C)  $6 x ^ { 2 } - 3 \sin 2 x - \frac { 3 } { 2 } \cos 2 x + C$ 
D)  $3 x ^ { 2 } + \frac { \sin ^ { 3 } x } { 3 } + C$

Question 6

Integrate the function.
-$$\int \frac { y ^ { 2 } } { \left( 49 - y ^ { 2 } \right) ^ { 3 / 2 } } d y$$

Accepted Answer

A)  $\frac { y } { \sqrt { 49 - y ^ { 2 } } } + C$ 
B)  $\frac { 7 y } { \sqrt { 49 - y ^ { 2 } } } - \sin ^ { - 1 } y + C$ 
C)  $\frac { y } { \sqrt { 49 - y ^ { 2 } } } - \sin ^ { - 1 } \left( \frac { y } { 7 } \right) + C$ 
D)  $\sqrt { 49 - y ^ { 2 } } - \sin ^ { - 1 } \left( \frac { y } { 7 } \right) + C$ 
A)  $\frac { y } { \sqrt { 49 - y ^ { 2 } } } + C$ 
B)  $\frac { 7 y } { \sqrt { 49 - y ^ { 2 } } } - \sin ^ { - 1 } y + C$ 
C)  $\frac { y } { \sqrt { 49 - y ^ { 2 } } } - \sin ^ { - 1 } \left( \frac { y } { 7 } \right) + C$ 
D)  $\sqrt { 49 - y ^ { 2 } } - \sin ^ { - 1 } \left( \frac { y } { 7 } \right) + C$

Question 7

Use any method to evaluate the integral.
-$$\int \frac { \cos x } { \sin ^ { 2 } x } d x$$

Accepted Answer

A)  $- \csc x + C$ 
B)  $- \sec x + C$ 
C)  $\sec x + C$ 
D)  $\csc x + C$ 
A)  $- \csc x + C$ 
B)  $- \sec x + C$ 
C)  $\sec x + C$ 
D)  $\csc x + C$

Question 8

Evaluate the integral.
-$$\int \mathrm { e } ^ { 3 x } \cos 2 \mathrm { xdx }$$

Accepted Answer

A)  $\frac { 1 } { 13 } \left[ 2 \mathrm { e } ^ { 3 \mathrm { x } } \sin 2 \mathrm { x } + 3 \cos 2 \mathrm { x } \right] + \mathrm { C }$ 
B)  $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x - 3 \cos 2 x ] + C$ 
C)  $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x + 3 \cos 2 x ] + C$ 
D)  $\frac { \mathrm { e } ^ { 3 x } } { 2 } [ \sin 2 x + \cos 2 x ] + C$ 
A)  $\frac { 1 } { 13 } \left[ 2 \mathrm { e } ^ { 3 \mathrm { x } } \sin 2 \mathrm { x } + 3 \cos 2 \mathrm { x } \right] + \mathrm { C }$ 
B)  $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x - 3 \cos 2 x ] + C$ 
C)  $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x + 3 \cos 2 x ] + C$ 
D)  $\frac { \mathrm { e } ^ { 3 x } } { 2 } [ \sin 2 x + \cos 2 x ] + C$

Question 9

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { - 2 x }$, and the line $x = 2$ about the $y$-axis.

Accepted Answer

A)  $\frac { 1 } { 2 } \pi \left( 1 - 5 \mathrm { e } ^ { - 4 } \right)$ 
B)  $\frac { 1 } { 2 } \pi \left( 1 - 3 \mathrm { e } ^ { - 4 } \right)$ 
C)  $\frac { 1 } { 2 } \pi \left( 1 - 4 \mathrm { e } ^ { - 4 } \right)$
D - $\frac { 1 } { 2 } \pi \left( 1 + 5 e ^ { - 4 } \right)$ 
A)  $\frac { 1 } { 2 } \pi \left( 1 - 5 \mathrm { e } ^ { - 4 } \right)$ 
B)  $\frac { 1 } { 2 } \pi \left( 1 - 3 \mathrm { e } ^ { - 4 } \right)$ 
C)  $\frac { 1 } { 2 } \pi \left( 1 - 4 \mathrm { e } ^ { - 4 } \right)$
D - $\frac { 1 } { 2 } \pi \left( 1 + 5 e ^ { - 4 } \right)$

Question 10

Evaluate the integral by using a substitution prior to integration by parts.
-$$\int \cos ( \ln x ) d x$$

Accepted Answer

A)  $\frac { x } { 2 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$ 
B)  $x \cos ( \ln x ) + \sin ( \ln x ) + C$ 
C)  $\frac { x } { 2 } [ \cos ( \ln x ) - \sin ( \ln x ) ] + C$ 
D)  $x [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$ 
A)  $\frac { x } { 2 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$ 
B)  $x \cos ( \ln x ) + \sin ( \ln x ) + C$ 
C)  $\frac { x } { 2 } [ \cos ( \ln x ) - \sin ( \ln x ) ] + C$ 
D)  $x [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$

Question 11

Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }$$

Accepted Answer

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

Question 12

Use integration by parts to establish a reduction formula for the integral.
-$$\int \cos ^ { n } x d x$$

Accepted Answer

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

Question 13

Evaluate the integral. The integral may not require integration by parts.
-$$\int x ^ { 7 } \sec x ^ { 8 } d x$$

Accepted Answer

A)  $8 \ln \left| \sec x ^ { 8 } - \tan x ^ { 8 } \right| + C$ 
B)  $x ^ { 9 } \sec x ^ { 9 } + C$ 
C)  $\frac { 1 } { 8 } \ln \left| \sec x ^ { 8 } + \tan x ^ { 8 } \right| + C$ 
D)  $\frac { 1 } { 7 } \ln \left| \sec x ^ { 8 } + \tan x ^ { 8 } \right| + C$ 
A)  $8 \ln \left| \sec x ^ { 8 } - \tan x ^ { 8 } \right| + C$ 
B)  $x ^ { 9 } \sec x ^ { 9 } + C$ 
C)  $\frac { 1 } { 8 } \ln \left| \sec x ^ { 8 } + \tan x ^ { 8 } \right| + C$ 
D)  $\frac { 1 } { 7 } \ln \left| \sec x ^ { 8 } + \tan x ^ { 8 } \right| + C$

Question 14

Integrate the function.
-$$\int \frac { 1 } { t ^ { 2 } \sqrt { 4 - t ^ { 2 } } } d t$$

Accepted Answer

A)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t } + C$ 
B)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t ^ { 2 } } + C$ 
C)  $\frac { \sqrt { 4 - t ^ { 2 } } } { t } + C$ 
D)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t } + \sin ^ { - 1 } 4 t + C$ 
A)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t } + C$ 
B)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t ^ { 2 } } + C$ 
C)  $\frac { \sqrt { 4 - t ^ { 2 } } } { t } + C$ 
D)  $- \frac { \sqrt { 4 - t ^ { 2 } } } { 4 t } + \sin ^ { - 1 } 4 t + C$

Question 15

Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { d x } { x \left( 1 + 4 \ln ^ { 2 } x \right) }$$

Accepted Answer

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

Question 16

Find the area of the region enclosed by the curve $y = x \sin x$ and the $x$-axis for $2 \pi \leq x \leq 3 \pi$.

Accepted Answer

A)  $4 \pi$ 
B)  0 
C)  $5 \pi$ 
D)  4 
A)  $4 \pi$ 
B)  0 
C)  $5 \pi$ 
D)  4

Question 17

Evaluate the integral.
-$$\int \left( x ^ { 2 } - 3 x \right) e ^ { x } d x$$

Accepted Answer

A)  $e ^ { x } \left[ x ^ { 2 } - 5 x - 5 \right] + C$ 
B)  $e ^ { x } \left[ x ^ { 2 } - 3 x + 3 \right] + C$ 
C)  $\frac { 1 } { 3 } x ^ { 3 } e ^ { x } - \frac { 3 } { 2 } x ^ { 2 } e ^ { x } + C$ 
D)  $e ^ { x } \left[ x ^ { 2 } - 5 x + 5 \right] + C$ 
A)  $e ^ { x } \left[ x ^ { 2 } - 5 x - 5 \right] + C$ 
B)  $e ^ { x } \left[ x ^ { 2 } - 3 x + 3 \right] + C$ 
C)  $\frac { 1 } { 3 } x ^ { 3 } e ^ { x } - \frac { 3 } { 2 } x ^ { 2 } e ^ { x } + C$ 
D)  $e ^ { x } \left[ x ^ { 2 } - 5 x + 5 \right] + C$

Question 18

Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { d x } { x \sqrt { x ^ { 2 } - 16 } }$$

Accepted Answer

A)  $\sin ^ { - 1 } \frac { x } { 4 } + C$ 
B)  $\frac { 1 } { 4 } \sin ^ { - 1 } \frac { x } { 4 } + C$ 
C)  $4 \sec ^ { - 1 } x + C$ 
D)  $\frac { 1 } { 4 } \sec ^ { - 1 } \left| \frac { x } { 4 } \right| + C$ 
A)  $\sin ^ { - 1 } \frac { x } { 4 } + C$ 
B)  $\frac { 1 } { 4 } \sin ^ { - 1 } \frac { x } { 4 } + C$ 
C)  $4 \sec ^ { - 1 } x + C$ 
D)  $\frac { 1 } { 4 } \sec ^ { - 1 } \left| \frac { x } { 4 } \right| + C$

Question 19

Solve the problem.
-Find the length of the curve $y = \ln ( \csc x ) , \pi / 4 \leq x \leq \pi / 2$

Accepted Answer

A)  $\ln ( \sqrt { 2 } + 1 )$ 
B)  $\ln ( 2 \sqrt { 2 } )$ 
C)  $\ln ( \sqrt { 2 } )$ 
D)  $1 - \ln ( \sqrt { 2 } + 1 )$ 
A)  $\ln ( \sqrt { 2 } + 1 )$ 
B)  $\ln ( 2 \sqrt { 2 } )$ 
C)  $\ln ( \sqrt { 2 } )$ 
D)  $1 - \ln ( \sqrt { 2 } + 1 )$

Question 20

Evaluate the integral by using a substitution prior to integration by parts.
-$$\int ( \ln 3 x ) ^ { 2 } d x$$

Accepted Answer

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

Integrate the function. - $\int _ { - 1 } ^ { 1 } \frac { 6 } { 1 + 36 t ^ { 2 } } d t$

Evaluate the integral. - $\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos 2 x } d x$

Use integration by parts to establish a reduction formula for the integral. - $\int x ^ { n } e ^ { - a x } d x$

Use various trigonometric identities to simplify the expression then integrate. - $\int \sin ^ { 2 } \theta \cos 5 \theta d \theta$

Use any method to evaluate the integral. - $\int 6 x \sin ^ { 2 } x d x$

Integrate the function. - $\int \frac { y ^ { 2 } } { \left( 49 - y ^ { 2 } \right) ^ { 3 / 2 } } d y$

Use any method to evaluate the integral. - $\int \frac { \cos x } { \sin ^ { 2 } x } d x$

Evaluate the integral. - $\int \mathrm { e } ^ { 3 x } \cos 2 \mathrm { xdx }$

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { - 2 x }$ , and the line $x = 2$ about the $y$ -axis.

Evaluate the integral by using a substitution prior to integration by parts. - $\int \cos ( \ln x ) d x$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }$

Use integration by parts to establish a reduction formula for the integral. - $\int \cos ^ { n } x d x$

Evaluate the integral. The integral may not require integration by parts. - $\int x ^ { 7 } \sec x ^ { 8 } d x$

Integrate the function. - $\int \frac { 1 } { t ^ { 2 } \sqrt { 4 - t ^ { 2 } } } d t$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { d x } { x \left( 1 + 4 \ln ^ { 2 } x \right) }$

Find the area of the region enclosed by the curve $y = x \sin x$ and the $x$ -axis for $2 \pi \leq x \leq 3 \pi$ .

Evaluate the integral. - $\int \left( x ^ { 2 } - 3 x \right) e ^ { x } d x$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { d x } { x \sqrt { x ^ { 2 } - 16 } }$

Solve the problem. -Find the length of the curve $y = \ln ( \csc x ) , \pi / 4 \leq x \leq \pi / 2$

Evaluate the integral by using a substitution prior to integration by parts. - $\int ( \ln 3 x ) ^ { 2 } d x$

Functions

Limits and Derivatives

Differentiation

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

First-Order Differential Equations

Infinite Sequences and Series

Filters

Exam 8: Techniques of Integration

Integrate the function. - ∫−1161+36t2dt\int _ { - 1 } ^ { 1 } \frac { 6 } { 1 + 36 t ^ { 2 } } d t∫−11​1+36t26​dt

Evaluate the integral. - ∫0π1−cos⁡2xdx\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos 2 x } d x∫0π​1−cos2x​dx

Use integration by parts to establish a reduction formula for the integral. - ∫xne−axdx\int x ^ { n } e ^ { - a x } d x∫xne−axdx

Use various trigonometric identities to simplify the expression then integrate. - ∫sin⁡2θcos⁡5θdθ\int \sin ^ { 2 } \theta \cos 5 \theta d \theta∫sin2θcos5θdθ

Use any method to evaluate the integral. - ∫6xsin⁡2xdx\int 6 x \sin ^ { 2 } x d x∫6xsin2xdx

Integrate the function. - ∫y2(49−y2)3/2dy\int \frac { y ^ { 2 } } { \left( 49 - y ^ { 2 } \right) ^ { 3 / 2 } } d y∫(49−y2)3/2y2​dy

Use any method to evaluate the integral. - ∫cos⁡xsin⁡2xdx\int \frac { \cos x } { \sin ^ { 2 } x } d x∫sin2xcosx​dx

Evaluate the integral. - ∫e3xcos⁡2xdx\int \mathrm { e } ^ { 3 x } \cos 2 \mathrm { xdx }∫e3xcos2xdx

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y=e−2xy = e ^ { - 2 x }y=e−2x , and the line x=2x = 2x=2 about the yyy -axis.

Evaluate the integral by using a substitution prior to integration by parts. - ∫cos⁡(ln⁡x)dx\int \cos ( \ln x ) d x∫cos(lnx)dx

Use a trigonometric substitution to evaluate the integral. - ∫exdx1−e2x\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }∫1−e2x​exdx​

Use integration by parts to establish a reduction formula for the integral. - ∫cos⁡nxdx\int \cos ^ { n } x d x∫cosnxdx

Evaluate the integral. The integral may not require integration by parts. - ∫x7sec⁡x8dx\int x ^ { 7 } \sec x ^ { 8 } d x∫x7secx8dx

Integrate the function. - ∫1t24−t2dt\int \frac { 1 } { t ^ { 2 } \sqrt { 4 - t ^ { 2 } } } d t∫t24−t2​1​dt

Use a trigonometric substitution to evaluate the integral. - ∫dxx(1+4ln⁡2x)\int \frac { d x } { x \left( 1 + 4 \ln ^ { 2 } x \right) }∫x(1+4ln2x)dx​

Find the area of the region enclosed by the curve y=xsin⁡xy = x \sin xy=xsinx and the xxx -axis for 2π≤x≤3π2 \pi \leq x \leq 3 \pi2π≤x≤3π .

Evaluate the integral. - ∫(x2−3x)exdx\int \left( x ^ { 2 } - 3 x \right) e ^ { x } d x∫(x2−3x)exdx

Use a trigonometric substitution to evaluate the integral. - ∫dxxx2−16\int \frac { d x } { x \sqrt { x ^ { 2 } - 16 } }∫xx2−16​dx​

Solve the problem. -Find the length of the curve y=ln⁡(csc⁡x),π/4≤x≤π/2y = \ln ( \csc x ) , \pi / 4 \leq x \leq \pi / 2y=ln(cscx),π/4≤x≤π/2

Evaluate the integral by using a substitution prior to integration by parts. - ∫(ln⁡3x)2dx\int ( \ln 3 x ) ^ { 2 } d x∫(ln3x)2dx