Question 1

Find the definite integral. $\int _ { 0 } ^ { \pi / 9 } x \cos 9 x d x$

Accepted Answer

A)  $\frac { 2 } { 81 }$ 
B)  $- \frac { 2 } { 81 }$ 
C)  $- \frac { 4 } { 9 }$ 
D)  $- \frac { 2 } { 9 }$ 
E)  $\frac { 4 } { 9 }$ 
A)  $\frac { 2 } { 81 }$ 
B)  $- \frac { 2 } { 81 }$ 
C)  $- \frac { 4 } { 9 }$ 
D)  $- \frac { 2 } { 9 }$ 
E)  $\frac { 4 } { 9 }$

Question 2

Use integration tables to evaluate the integral $\int _ { 0 } ^ { \pi / 2 } 21 x \cos x d x$

Accepted Answer

A)  $21 \pi$ 
B)  $\frac { 21 \pi } { 2 }$ 
C)  $21 \left( \frac { \pi - 1 } { 2 } \right)$ 
D)  $21 ( \pi - 1 )$ 
E)  $21 \left( \frac { \pi } { 2 } - 1 \right)$ 
A)  $21 \pi$ 
B)  $\frac { 21 \pi } { 2 }$ 
C)  $21 \left( \frac { \pi - 1 } { 2 } \right)$ 
D)  $21 ( \pi - 1 )$ 
E)  $21 \left( \frac { \pi } { 2 } - 1 \right)$

Question 3

Solve the differential equation
$\left( 36 + \tan ^ { 2 } x \right) y ^ {t } = \sec ^ { 2 } x$

Accepted Answer

A)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \sec x } { 6 } \right) + C$ 
B)  $y = \arctan \left( \frac { \sec x } { 6 } \right) + C$ 
C)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \tan x \sec x } { 6 } \right) + C$ 
D)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \tan x } { 6 } \right) + C$ 
E) $y = \arctan \left( \frac { \tan x } { 6 } \right) + C$ 
A)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \sec x } { 6 } \right) + C$ 
B)  $y = \arctan \left( \frac { \sec x } { 6 } \right) + C$ 
C)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \tan x \sec x } { 6 } \right) + C$ 
D)  $y = \frac { 1 } { 6 } \arctan \left( \frac { \tan x } { 6 } \right) + C$ 
E) $y = \arctan \left( \frac { \tan x } { 6 } \right) + C$

Question 4

Find the indefinite integral. $$\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x$$

Accepted Answer

A) $- \frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
B) $\frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
C) $- \frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
D) $\frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
E)  $8 x \sqrt { x ^ { 2 } + 8 } + C$ 
A) $- \frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
B) $\frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
C) $- \frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
D) $\frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$ 
E)  $8 x \sqrt { x ^ { 2 } + 8 } + C$

Question 5

Determine whether the improper integral $\int _ { - \infty } ^ { \infty } \frac { 9 } { 49 + x ^ { 2 } } d x$ diverges or converges. Evaluate the integral if it converges.

Accepted Answer

A)  $\frac { 9 } { 49 }$ 
B)  $\frac { 9 } { 7 } \pi$ 
C)  $\frac { 9 } { 49 } \pi$ 
D)  $\frac { 9 } { 7 }$ 
E)  diverges 
A)  $\frac { 9 } { 49 }$ 
B)  $\frac { 9 } { 7 } \pi$ 
C)  $\frac { 9 } { 49 } \pi$ 
D)  $\frac { 9 } { 7 }$ 
E)  diverges

Question 6

$\text { Find the indefinite integral by making the substitution } x = 6 \sec \theta \text {. }$ $$\int \frac { 1 } { \sqrt { x ^ { 2 } - 36 } } d x$$

Accepted Answer

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

Question 7

$\text { Find the indefinite integral } \int 4 \csc ^ { 2 } x e ^ { \cot x } d x \text {. }$

Accepted Answer

A)  $- 4 e ^ { \cot x } + C$ 
B)  $- 4 \cot x \csc x + C$ 
C)  $- 4 \cot x + C$ 
D)  $e ^ { 5 \cot x } + C$ 
E)  $- 4 \csc x + C$ 
A)  $- 4 e ^ { \cot x } + C$ 
B)  $- 4 \cot x \csc x + C$ 
C)  $- 4 \cot x + C$ 
D)  $e ^ { 5 \cot x } + C$ 
E)  $- 4 \csc x + C$

Question 8

$\text { Find } \int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x$

Accepted Answer

A)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 8 } \sin \left( x ^ { 8 } \right) - \cos \left( x ^ { 8 } \right) + C$ 
B)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 9 } \sin \left( x ^ { 9 } \right) + \cos \left( x ^ { 9 } \right) + C$ 
C)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 8 } \sin \left( x ^ { 8 } \right) + \cos \left( x ^ { 8 } \right) + C$ 
D)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 15 } \sin \left( x ^ { 15 } \right) - \cos \left( x ^ { 15 } \right) + C$ 
E)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 15 } \sin \left( x ^ { 15 } \right) + \cos \left( x ^ { 15 } \right) + C$ 
A)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 8 } \sin \left( x ^ { 8 } \right) - \cos \left( x ^ { 8 } \right) + C$ 
B)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 9 } \sin \left( x ^ { 9 } \right) + \cos \left( x ^ { 9 } \right) + C$ 
C)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 8 } \sin \left( x ^ { 8 } \right) + \cos \left( x ^ { 8 } \right) + C$ 
D)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 15 } \sin \left( x ^ { 15 } \right) - \cos \left( x ^ { 15 } \right) + C$ 
E)  $\int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x = x ^ { 15 } \sin \left( x ^ { 15 } \right) + \cos \left( x ^ { 15 } \right) + C$

Question 9

$\text { Find } \int \sin 5 x \cos 10 x d x \text {. }$

Accepted Answer

A)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 30 } ( 3 \cos 5 x - \cos 15 x ) + C$ 
B)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 15 } ( 3 \cos 14 x + 3 \cos 5 x ) + C$ 
C)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 31 } ( \cos 15 x - 3 \cos 5 x ) + C$ 
D)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 30 } ( 4 \cos 5 x + \cos 14 x ) + C$ 
E)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 31 } ( 3 \cos 5 x - \cos 15 x ) + C$ 
A)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 30 } ( 3 \cos 5 x - \cos 15 x ) + C$ 
B)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 15 } ( 3 \cos 14 x + 3 \cos 5 x ) + C$ 
C)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 31 } ( \cos 15 x - 3 \cos 5 x ) + C$ 
D)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 30 } ( 4 \cos 5 x + \cos 14 x ) + C$ 
E)  $\int \sin 5 x \cos 10 x d x = \frac { 1 } { 31 } ( 3 \cos 5 x - \cos 15 x ) + C$

Question 10

$\text { Use partial fractions to find } \int \frac { 1 } { x ^ { 2 } - 9 } d x \text {. }$

Accepted Answer

A)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 } \ln \left| \frac { x - 3 } { x + 3 } \right| + C$ 
B)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 ( x - 3 ) } - \frac { 1 } { 6 ( x + 3 ) } + C$ 
C)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 } \ln \left| \frac { x - 6 } { x + 6 } \right| + C$ 
D)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { ( x - 3 ) } - \frac { 1 } { ( x + 3 ) } + C$ 
E) $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \ln \left| \frac { x - 3 } { x + 3 } \right| + C$ 
A)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 } \ln \left| \frac { x - 3 } { x + 3 } \right| + C$ 
B)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 ( x - 3 ) } - \frac { 1 } { 6 ( x + 3 ) } + C$ 
C)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { 6 } \ln \left| \frac { x - 6 } { x + 6 } \right| + C$ 
D)  $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \frac { 1 } { ( x - 3 ) } - \frac { 1 } { ( x + 3 ) } + C$ 
E) $\int \frac { 1 } { x ^ { 2 } - 9 } d x = \ln \left| \frac { x - 3 } { x + 3 } \right| + C$

Question 11

Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by $y = e ^ { - 8 t } ( \cos 2 t + 9 \sin 2 t )$. Find the average value of $y$ on the interval from $t = 0$ to $t = \pi$. Round your answer to three decimal places.

Accepted Answer

A)  $0.406$ 
B)  $0.131$ 
C)  $0.122$ 
D)  $0.129$ 
E)  $0.382$ 
A)  $0.406$ 
B)  $0.131$ 
C)  $0.122$ 
D)  $0.129$ 
E)  $0.382$

Question 12

Find the indefinite integral. $$\int \sin x \cos ^ { 3 } x d x$$

Accepted Answer

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

Question 13

Find the indefinite integral. $$\int \tan ^ { 3 } 9 x d x$$

Accepted Answer

A)  $\frac { 1 } { 18 } \tan ^ { 2 } 9 x - \frac { 1 } { 9 } \ln | \cos 9 x | + C$ 
B)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \tan ^ { 2 } 9 x \sec 9 x + C$ 
C)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \tan ^ { 2 } 9 x + C$ 
D)  $\frac { 1 } { 9 } \ln | \cos 9 x | - \frac { 1 } { 18 } \sec ^ { 2 } 9 x + C$ 
E)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \sec ^ { 2 } 9 x + C$ 
A)  $\frac { 1 } { 18 } \tan ^ { 2 } 9 x - \frac { 1 } { 9 } \ln | \cos 9 x | + C$ 
B)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \tan ^ { 2 } 9 x \sec 9 x + C$ 
C)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \tan ^ { 2 } 9 x + C$ 
D)  $\frac { 1 } { 9 } \ln | \cos 9 x | - \frac { 1 } { 18 } \sec ^ { 2 } 9 x + C$ 
E)  $\frac { 1 } { 9 } \ln | \cos 9 x | + \frac { 1 } { 18 } \sec ^ { 2 } 9 x + C$

Question 14

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 4 x ^ { 2 } + 3 } { ( x + 8 ) ^ { 3 } }$

Accepted Answer

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

Question 15

Find the indefinite integral. 
$\int e ^ { t } \sin 6 t d t$

Accepted Answer

A)  $\left( \frac { \sin 6 t + 6 \cos 6 t } { 37 } \right) e ^ { t } + C$ 
B)  $\left( \frac { - 6 \sin 6 t + \cos 6 t } { 37 } \right) e ^ { t } + C$ 
C)  $\left( \frac { 6 \sin 6 t + \cos 6 t } { 37 } \right) e ^ { t } + C$ 
D)  $\left( \frac { - \sin 6 t + 6 \cos 6 t } { 37 } \right) e ^ { t } + C$ 
E)  $\left( \frac { \sin 6 t - 6 \cos 6 t } { 37 } \right) e ^ { t } + C$ 
A)  $\left( \frac { \sin 6 t + 6 \cos 6 t } { 37 } \right) e ^ { t } + C$ 
B)  $\left( \frac { - 6 \sin 6 t + \cos 6 t } { 37 } \right) e ^ { t } + C$ 
C)  $\left( \frac { 6 \sin 6 t + \cos 6 t } { 37 } \right) e ^ { t } + C$ 
D)  $\left( \frac { - \sin 6 t + 6 \cos 6 t } { 37 } \right) e ^ { t } + C$ 
E)  $\left( \frac { \sin 6 t - 6 \cos 6 t } { 37 } \right) e ^ { t } + C$

Question 16

$\text { Evaluate the definite integral } \int \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x \text {. }$

Accepted Answer

A)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = 5$ 
B)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 3 )$ 
C)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = 2$ 
D)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 4 )$ 
E)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 1 )$ 
A)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = 5$ 
B)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 3 )$ 
C)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = 2$ 
D)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 4 )$ 
E)  $\int _ { 0 } ^ { 2 } \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x = \ln ( 1 )$

Question 17

$$\text { Evaluate the integral } \int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \text { using the integration limits. Round your answer }$$ to three decimal places.

Accepted Answer

A)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 10.728$ 
B)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 5.272$ 
C)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 9$ 
D)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 9.392$ 
E)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 7.608$ 
A)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 10.728$ 
B)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 5.272$ 
C)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 9$ 
D)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 9.392$ 
E)  $\int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \approx 7.608$

Question 18

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \frac { \ln \left( x ^ { 2 } \right) } { x ^ { 7 } } \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C)  $\frac { 2 } { 7 }$ 
D)  0 
E)  $\frac { 7 } { 2 }$ 
A)  $- \infty$ 
B)  $\infty$ 
C)  $\frac { 2 } { 7 }$ 
D)  0 
E)  $\frac { 7 } { 2 }$

Question 19

Find the indefinite integral. 
$\int e ^ { 2 v } \cos 7 v d v$

Accepted Answer

A)  $\left( \frac { 2 \cos 7 v + 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$ 
B)  $\left( \frac { - 2 \cos 7 v - 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$ 
C)  $\left( \frac { - 2 \cos 7 v + 7 \sin 7 v } { 106 } \right) e ^ { 2 v } + C$ 
D)  $\left( \frac { 2 \cos 7 v - 7 \sin 7 v } { 106 } \right) e ^ { 2 v } + C$ 
E)  $\left( \frac { 2 \cos 7 v - 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$ 
A)  $\left( \frac { 2 \cos 7 v + 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$ 
B)  $\left( \frac { - 2 \cos 7 v - 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$ 
C)  $\left( \frac { - 2 \cos 7 v + 7 \sin 7 v } { 106 } \right) e ^ { 2 v } + C$ 
D)  $\left( \frac { 2 \cos 7 v - 7 \sin 7 v } { 106 } \right) e ^ { 2 v } + C$ 
E)  $\left( \frac { 2 \cos 7 v - 7 \sin 7 v } { 53 } \right) e ^ { 2 v } + C$

Question 20

$$\text { Evaluate } \int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \text {. Round your answer to three decimal places. }$$

Accepted Answer

A)  $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 3.397$ 
B)  $\int _ { 0 } ^ { 4 x } e ^ { 4 x } \sin x d x \approx 12.604$ 
C)   $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 9.134$ 
D) $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 43.054$ 
E) $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 31.254$ 
A)  $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 3.397$ 
B)  $\int _ { 0 } ^ { 4 x } e ^ { 4 x } \sin x d x \approx 12.604$ 
C)   $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 9.134$ 
D) $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 43.054$ 
E) $\int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \approx 31.254$

Find the definite integral. $\int _ { 0 } ^ { \pi / 9 } x \cos 9 x d x$

Use integration tables to evaluate the integral $\int _ { 0 } ^ { \pi / 2 } 21 x \cos x d x$

Solve the differential equation $\left( 36 + \tan ^ { 2 } x \right) y ^ {t } = \sec ^ { 2 } x$

Find the indefinite integral. $\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x$

Determine whether the improper integral $\int _ { - \infty } ^ { \infty } \frac { 9 } { 49 + x ^ { 2 } } d x$ diverges or converges. Evaluate the integral if it converges.

$\text { Find the indefinite integral by making the substitution } x = 6 \sec \theta \text {. }$ $\int \frac { 1 } { \sqrt { x ^ { 2 } - 36 } } d x$

$\text { Find the indefinite integral } \int 4 \csc ^ { 2 } x e ^ { \cot x } d x \text {. }$

$\text { Find } \int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x$

$\text { Find } \int \sin 5 x \cos 10 x d x \text {. }$

$\text { Use partial fractions to find } \int \frac { 1 } { x ^ { 2 } - 9 } d x \text {. }$

Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by $y = e ^ { - 8 t } ( \cos 2 t + 9 \sin 2 t )$ . Find the average value of $y$ on the interval from $t = 0$ to $t = \pi$ . Round your answer to three decimal places.

Find the indefinite integral. $\int \sin x \cos ^ { 3 } x d x$

Find the indefinite integral. $\int \tan ^ { 3 } 9 x d x$

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 4 x ^ { 2 } + 3 } { ( x + 8 ) ^ { 3 } }$

Find the indefinite integral. $\int e ^ { t } \sin 6 t d t$

$\text { Evaluate the definite integral } \int \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x \text {. }$

$\text { Evaluate the integral } \int _ { 0 } ^ { 3 } \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 9 } } d x \text { using the integration limits. Round your answer }$ to three decimal places.

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \frac { \ln \left( x ^ { 2 } \right) } { x ^ { 7 } } \text { using L'Hopital's Rule if necessary. }$

Find the indefinite integral. $\int e ^ { 2 v } \cos 7 v d v$

$\text { Evaluate } \int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \text {. Round your answer to three decimal places. }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 8: Basic Integration Rules

Find the definite integral. ∫0π/9xcos⁡9xdx\int _ { 0 } ^ { \pi / 9 } x \cos 9 x d x∫0π/9​xcos9xdx

Use integration tables to evaluate the integral ∫0π/221xcos⁡xdx\int _ { 0 } ^ { \pi / 2 } 21 x \cos x d x∫0π/2​21xcosxdx

Solve the differential equation (36+tan⁡2x)yt=sec⁡2x\left( 36 + \tan ^ { 2 } x \right) y ^ {t } = \sec ^ { 2 } x(36+tan2x)yt=sec2x

Find the indefinite integral. ∫1(x2+8)3/2dx\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x∫(x2+8)3/21​dx

Determine whether the improper integral ∫−∞∞949+x2dx\int _ { - \infty } ^ { \infty } \frac { 9 } { 49 + x ^ { 2 } } d x∫−∞∞​49+x29​dx diverges or converges. Evaluate the integral if it converges.

Find the indefinite integral ∫4csc⁡2xecot⁡xdx. \text { Find the indefinite integral } \int 4 \csc ^ { 2 } x e ^ { \cot x } d x \text {. } Find the indefinite integral ∫4csc2xecotxdx.

Find ∫8x15cos⁡(x8)dx\text { Find } \int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x Find ∫8x15cos(x8)dx

Find ∫sin⁡5xcos⁡10xdx. \text { Find } \int \sin 5 x \cos 10 x d x \text {. } Find ∫sin5xcos10xdx.

Use partial fractions to find ∫1x2−9dx. \text { Use partial fractions to find } \int \frac { 1 } { x ^ { 2 } - 9 } d x \text {. } Use partial fractions to find ∫x2−91​dx.

Find the indefinite integral. ∫sin⁡xcos⁡3xdx\int \sin x \cos ^ { 3 } x d x∫sinxcos3xdx

Find the indefinite integral. ∫tan⁡39xdx\int \tan ^ { 3 } 9 x d x∫tan39xdx

Write the form of the partial fraction decomposition for the following rational expression. 4x2+3(x+8)3\frac { 4 x ^ { 2 } + 3 } { ( x + 8 ) ^ { 3 } }(x+8)34x2+3​

Find the indefinite integral. ∫etsin⁡6tdt\int e ^ { t } \sin 6 t d t∫etsin6tdt

Evaluate the definite integral ∫34x2+5x+1dx. \text { Evaluate the definite integral } \int \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x \text {. } Evaluate the definite integral ∫4x2+5x+13​dx.

Find the indefinite integral. ∫e2vcos⁡7vdv\int e ^ { 2 v } \cos 7 v d v∫e2vcos7vdv

Evaluate ∫01e4xsin⁡xdx. Round your answer to three decimal places. \text { Evaluate } \int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \text {. Round your answer to three decimal places. } Evaluate ∫01​e4xsinxdx. Round your answer to three decimal places.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

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Find the definite integral. $\int _ { 0 } ^ { \pi / 9 } x \cos 9 x d x$

Use integration tables to evaluate the integral $\int _ { 0 } ^ { \pi / 2 } 21 x \cos x d x$

Solve the differential equation $\left( 36 + \tan ^ { 2 } x \right) y ^ {t } = \sec ^ { 2 } x$

Find the indefinite integral. $\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x$

Determine whether the improper integral $\int _ { - \infty } ^ { \infty } \frac { 9 } { 49 + x ^ { 2 } } d x$ diverges or converges. Evaluate the integral if it converges.

$\text { Find the indefinite integral } \int 4 \csc ^ { 2 } x e ^ { \cot x } d x \text {. }$

$\text { Find } \int 8 x ^ { 15 } \cos \left( x ^ { 8 } \right) d x$

$\text { Find } \int \sin 5 x \cos 10 x d x \text {. }$

$\text { Use partial fractions to find } \int \frac { 1 } { x ^ { 2 } - 9 } d x \text {. }$

Find the indefinite integral. $\int \sin x \cos ^ { 3 } x d x$

Find the indefinite integral. $\int \tan ^ { 3 } 9 x d x$

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 4 x ^ { 2 } + 3 } { ( x + 8 ) ^ { 3 } }$

Find the indefinite integral. $\int e ^ { t } \sin 6 t d t$

$\text { Evaluate the definite integral } \int \frac { 3 } { 4 x ^ { 2 } + 5 x + 1 } d x \text {. }$

Find the indefinite integral. $\int e ^ { 2 v } \cos 7 v d v$

$\text { Evaluate } \int _ { 0 } ^ { 1 } e ^ { 4 x } \sin x d x \text {. Round your answer to three decimal places. }$