Question 1

$\text { Find the indefinite integral } \int 3 x ^ { 4 } \cos 8 \pi x ^ { 5 } d x \text {. }$

Accepted Answer

A)  $\frac { 3 } { 40 \pi } \sec 40 x ^ { 4 } + C$ 
B)  $\frac { 3 } { 40 \pi } \sin 40 \pi \pi ^ { 4 } + C$ 
C)  $\frac { 3 } { 40 \pi } \sec 120 \pi x ^ { 5 } + C$ 
D)  $\frac { 3 } { 40 \pi } \tan 8 \pi x ^ { 4 } + C$ 
E) $\frac { 3 } { 40 \pi } \sin 8 \pi x ^ { 5 } + C$ 
A)  $\frac { 3 } { 40 \pi } \sec 40 x ^ { 4 } + C$ 
B)  $\frac { 3 } { 40 \pi } \sin 40 \pi \pi ^ { 4 } + C$ 
C)  $\frac { 3 } { 40 \pi } \sec 120 \pi x ^ { 5 } + C$ 
D)  $\frac { 3 } { 40 \pi } \tan 8 \pi x ^ { 4 } + C$ 
E) $\frac { 3 } { 40 \pi } \sin 8 \pi x ^ { 5 } + C$ E

Question 2

Find the indefinite integral. $$\int \frac { 3 m } { m - 9 } d m$$

Accepted Answer

A)  $27 \ln | m - 9 | - 3 m + C$ 
B)  $- 27 \ln | m - 9 | + 3 m + C$ 
C)  $27 \ln | m - 9 | + 3 m ^ { 2 } + C$ 
D)  $27 \ln | m - 9 | + 3 m + C$ 
E)  $- 27 \ln | m - 9 | + 3 m ^ { 2 } + C$ 
A)  $27 \ln | m - 9 | - 3 m + C$ 
B)  $- 27 \ln | m - 9 | + 3 m + C$ 
C)  $27 \ln | m - 9 | + 3 m ^ { 2 } + C$ 
D)  $27 \ln | m - 9 | + 3 m + C$ 
E)  $- 27 \ln | m - 9 | + 3 m ^ { 2 } + C$ D

Question 3

$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$

Accepted Answer

A)  $2 \arcsin 15$ 
B)  $\frac { 2 } { \sqrt { 15 } } \arctan \sqrt { 15 }$ 
C)  $\frac { 1 } { \sqrt { 15 } } \arcsin 15$ 
D)  $15 \arctan 15$ 
E)  $15 \arcsin \sqrt { 15 }$ 
A)  $2 \arcsin 15$ 
B)  $\frac { 2 } { \sqrt { 15 } } \arctan \sqrt { 15 }$ 
C)  $\frac { 1 } { \sqrt { 15 } } \arcsin 15$ 
D)  $15 \arctan 15$ 
E)  $15 \arcsin \sqrt { 15 }$ B

Question 4

$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } \frac { x ^ { 5 } } { e ^ { 4 x } } \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  0 
B)  $\frac { 15 } { 128 }$ 
C)  $\frac { 5 } { 4 }$ 
D)  1 
E)  does not exist 
A)  0 
B)  $\frac { 15 } { 128 }$ 
C)  $\frac { 5 } { 4 }$ 
D)  1 
E)  does not exist

Question 5

$\text { Determine whether the improper integral } \int _ { 0 } ^ { 2 } \frac { 3 } { x } d x \text { diverges or converges. Evaluate the }$ integral if it converges.

Accepted Answer

A)  3 
B)  2 
C)  1 
D)  $\frac { 2 } { 3 }$ 
E)  diverges 
A)  3 
B)  2 
C)  1 
D)  $\frac { 2 } { 3 }$ 
E)  diverges

Question 6

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \pi / 4 } 3 \tan \theta d \theta \text { diverges or converges. }$

Accepted Answer

A)  diverges 
B)  converges 
A)  diverges 
B)  converges

Question 7

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

Accepted Answer

A)  $\frac { \sin 3 x \left( 3 - \sin ^ { 2 } 3 x \right) } { 9 } + C$ 
B)  $\frac { \sin 3 x \left( 3 - \sin ^ { 4 } 3 x \right) } { 9 } + C$ 
C)  $\frac { \sin 3 x \left( 3 + \sin ^ { 2 } 3 x \right) } { 3 } + C$ 
D)  $\frac { \sin 3 x \left( 3 + \sin ^ { 4 } 3 x \right) } { 9 } + C$ 
E)  $\frac { \sin 3 x \left( 3 + \sin ^ { 2 } 3 x \right) } { 9 } + C$ 
A)  $\frac { \sin 3 x \left( 3 - \sin ^ { 2 } 3 x \right) } { 9 } + C$ 
B)  $\frac { \sin 3 x \left( 3 - \sin ^ { 4 } 3 x \right) } { 9 } + C$ 
C)  $\frac { \sin 3 x \left( 3 + \sin ^ { 2 } 3 x \right) } { 3 } + C$ 
D)  $\frac { \sin 3 x \left( 3 + \sin ^ { 4 } 3 x \right) } { 9 } + C$ 
E)  $\frac { \sin 3 x \left( 3 + \sin ^ { 2 } 3 x \right) } { 9 } + C$

Question 8

$\text { Use partial fractions to find } \int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 3 } + 3 x ^ { 2 } } d x$

Accepted Answer

A) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \ln \left| x ^ { 6 } + 3 x ^ { 5 } \right| + C$ 
B) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \left( x ^ { 6 } + 3 x ^ { 5 } \right) + C$ 
C) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \ln \left| x ^ { 7 } + x ^ { 5 } \right| + C$ 
D) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \ln \left| x ^ { 6 } + 5 x ^ { 4 } \right| + C$ 
E) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \ln \left| x ^ { 6 } + 3 x ^ { 5 } \right| + C$ 
A) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \ln \left| x ^ { 6 } + 3 x ^ { 5 } \right| + C$ 
B) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \left( x ^ { 6 } + 3 x ^ { 5 } \right) + C$ 
C) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \ln \left| x ^ { 7 } + x ^ { 5 } \right| + C$ 
D) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \ln \left| x ^ { 6 } + 5 x ^ { 4 } \right| + C$ 
E) $\int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 2 } ( x + 3 ) } d x = \frac { 1 } { x } + \ln \left| x ^ { 6 } + 3 x ^ { 5 } \right| + C$

Question 9

Find the indefinite integral by making the substitution 
$x = 6 \tan \theta$ $\int x \sqrt { 36 + x ^ { 2 } } d x$

Accepted Answer

A)  $- \frac { 2 \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
B)  $- \frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
C)  $\frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
D) $\frac { 2 \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
E) $\frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 2 } + C$ 
A)  $- \frac { 2 \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
B)  $- \frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
C)  $\frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
D) $\frac { 2 \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 3 } + C$ 
E) $\frac { \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } } { 2 } + C$

Question 10

$\text { Use integration tables to find } \int \frac { 1 } { 5 x ^ { 2 } + 3 x + 3 } d x \text {. }$

Accepted Answer

A)  $\frac { 2 } { \sqrt { 51 } } \arctan \left( \frac { 10 x + 3 } { \sqrt { 51 } } \right) + C$ 
B)  $\quad \frac { 1 } { \sqrt { 51 } } \ln \left| \frac { 10 x + 3 - \sqrt { 51 } } { 10 x + 3 + \sqrt { 51 } } \right| + C$ 
C)  $\frac { 1 } { 10 } \left( \ln \left| 3 + 3 x + 5 x ^ { 2 } \right| - \frac { 6 } { \sqrt { 51 } } \arctan \left( \frac { 10 x + 3 } { \sqrt { 51 } } \right) \right) + C$ 
D)  $\frac { 1 } { \sqrt { 51 } } \ln \left| \frac { 10 x - 3 + \sqrt { 51 } } { 10 x + 3 + \sqrt { 51 } } \right| + C$ 
E)  $\frac { 2 } { \sqrt { 51 } } \arctan \left( \frac { 10 x - 3 } { \sqrt { 51 } } \right) + C$ 
A)  $\frac { 2 } { \sqrt { 51 } } \arctan \left( \frac { 10 x + 3 } { \sqrt { 51 } } \right) + C$ 
B)  $\quad \frac { 1 } { \sqrt { 51 } } \ln \left| \frac { 10 x + 3 - \sqrt { 51 } } { 10 x + 3 + \sqrt { 51 } } \right| + C$ 
C)  $\frac { 1 } { 10 } \left( \ln \left| 3 + 3 x + 5 x ^ { 2 } \right| - \frac { 6 } { \sqrt { 51 } } \arctan \left( \frac { 10 x + 3 } { \sqrt { 51 } } \right) \right) + C$ 
D)  $\frac { 1 } { \sqrt { 51 } } \ln \left| \frac { 10 x - 3 + \sqrt { 51 } } { 10 x + 3 + \sqrt { 51 } } \right| + C$ 
E)  $\frac { 2 } { \sqrt { 51 } } \arctan \left( \frac { 10 x - 3 } { \sqrt { 51 } } \right) + C$

Question 11

Find the indefinite integral by making the substitution $x = 7 \tan \theta$  $$\int \frac { x ^ { 3 } } { \sqrt { 49 + x ^ { 2 } } } d x$$

Accepted Answer

A)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 49 \right) } { 3 } + C$ 
B)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( 98 - x ^ { 2 } \right) } { 3 } + C$ 
C)  $\frac { 2 \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 49 \right) } { 3 } + C$ 
D)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 98 \right) } { 3 } + C$ 
E) $\frac { 2 \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 98 \right) } { 3 } + C$ 
A)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 49 \right) } { 3 } + C$ 
B)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( 98 - x ^ { 2 } \right) } { 3 } + C$ 
C)  $\frac { 2 \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 49 \right) } { 3 } + C$ 
D)  $\frac { \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 98 \right) } { 3 } + C$ 
E) $\frac { 2 \sqrt { 49 + x ^ { 2 } } \left( x ^ { 2 } - 98 \right) } { 3 } + C$

Question 12

Use substitution and partial fractions to find the indefinite integral. 
$\int \frac { \sqrt { x } } { x - 25 } d x$

Accepted Answer

A) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
B) 
$\int \frac { \sqrt { x } } { x - 25 } d x = \sqrt { x } + \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
C) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + \log ( 1 - \sqrt { x } ) + C$ 
D) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + x ^ { 2 } + 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
E) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
A) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
B) 
$\int \frac { \sqrt { x } } { x - 25 } d x = \sqrt { x } + \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
C) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + \log ( 1 - \sqrt { x } ) + C$ 
D) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + x ^ { 2 } + 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$ 
E) 
$\int \frac { \sqrt { x } } { x - 25 } d x = 2 \sqrt { x } + 5 \log ( 5 - \sqrt { x } ) - 4 \log ( \sqrt { x } - 5 ) + C$

Question 13

$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$

Accepted Answer

A)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
B)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } + \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
C)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$ 
D)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
E)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$ 
A)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
B)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } + \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
C)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$ 
D)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$ 
E)  $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$

Question 14

$\text { A population is growing according to the logistic model } N = \frac { 6000 } { 1 + e ^ { 4.2 - 1.6 ^ { t } } } \text {, where } t$ is the time in days. Find the average population over the interval $[ 0,2 ]$. Round your answer to one decimal place.

Accepted Answer

A)  $559.5 \mathrm { ft } . \mathrm { lbs }$ 
B)  $2434.5 \mathrm { ft }$ lbs 
C)  $716.1 \mathrm { ft } \mathrm { lbs }$ 
D)  $5117.6 \mathrm { ft } . \mathrm { lbs }$ 
E)  $3116.1 \mathrm { ft }$ lbs 
A)  $559.5 \mathrm { ft } . \mathrm { lbs }$ 
B)  $2434.5 \mathrm { ft }$ lbs 
C)  $716.1 \mathrm { ft } \mathrm { lbs }$ 
D)  $5117.6 \mathrm { ft } . \mathrm { lbs }$ 
E)  $3116.1 \mathrm { ft }$ lbs

Question 15

$\text { Apply the Extended Mean Value Theorem to the functions } f ( x ) = \frac { 4 } { x } \text { and }$ $g ( x ) = x ^ { 2 } - 64$ on the interval $[ 4,8 ]$, and find all values $c$ in the interval $( 4,8 )$ such that $\frac { f ^ { t } ( c ) } { g ^ {t } ( c ) } = \frac { f ( 8 ) - f ( 4 ) } { g ( 8 ) - g ( 4 ) }$

Accepted Answer

A)  $4 \sqrt [ 3 ] { 2 }$ 
B)  $2 \sqrt [ 3 ] { 12 }$ 
C)  $2 \sqrt { 6 }$ 
D)  $4 \sqrt [ 3 ] { 3 }$ 
E)  $8 \sqrt { 3 }$ 
A)  $4 \sqrt [ 3 ] { 2 }$ 
B)  $2 \sqrt [ 3 ] { 12 }$ 
C)  $2 \sqrt { 6 }$ 
D)  $4 \sqrt [ 3 ] { 3 }$ 
E)  $8 \sqrt { 3 }$

Question 16

Find the indefinite integral. $$\int \sec ^ { 4 } 9 x d x$$

Accepted Answer

A)  $\frac { 1 } { 27 } \tan ( 9 x ) ( 3 + \tan ( 9 x ) ) + C$ 
B)  $\frac { 1 } { 27 } \tan ( 9 x ) \left( 3 + \tan ^ { 2 } ( 9 x ) \right) + C$ 
C)  $- \frac { 1 } { 27 } \tan ( 9 x ) \left( 3 + \tan ^ { 2 } ( 9 x ) \right) + C$ 
D)  $\frac { 1 } { 27 } \tan ( 9 x ) \left( 3 - \tan ^ { 2 } ( 9 x ) \right) + C$ 
E)  $\frac { 1 } { 27 } \tan ( 9 x ) ( 3 - \tan ( 9 x ) ) + C$ 
A)  $\frac { 1 } { 27 } \tan ( 9 x ) ( 3 + \tan ( 9 x ) ) + C$ 
B)  $\frac { 1 } { 27 } \tan ( 9 x ) \left( 3 + \tan ^ { 2 } ( 9 x ) \right) + C$ 
C)  $- \frac { 1 } { 27 } \tan ( 9 x ) \left( 3 + \tan ^ { 2 } ( 9 x ) \right) + C$ 
D)  $\frac { 1 } { 27 } \tan ( 9 x ) \left( 3 - \tan ^ { 2 } ( 9 x ) \right) + C$ 
E)  $\frac { 1 } { 27 } \tan ( 9 x ) ( 3 - \tan ( 9 x ) ) + C$

Question 17

$\text { Find the indefinite integral } \int \frac { 1 } { \sqrt { 32 - 2 x - x ^ { 2 } } } d x \text {. }$

Accepted Answer

A) $\arcsin \left( \frac { x + 1 } { \sqrt { 32 } } \right) + C$ 
B)  $\arcsin \left( \frac { x + 2 } { \sqrt { 33 } } \right) + C$ 
C)  $\arcsin \left( \frac { x - 1 } { \sqrt { 32 } } \right) + C$ 
D)  $\arcsin \left( \frac { x - 2 } { \sqrt { 34 } } \right) + C$ 
E)  $\arcsin \left( \frac { x + 1 } { \sqrt { 33 } } \right) + C$ 
A) $\arcsin \left( \frac { x + 1 } { \sqrt { 32 } } \right) + C$ 
B)  $\arcsin \left( \frac { x + 2 } { \sqrt { 33 } } \right) + C$ 
C)  $\arcsin \left( \frac { x - 1 } { \sqrt { 32 } } \right) + C$ 
D)  $\arcsin \left( \frac { x - 2 } { \sqrt { 34 } } \right) + C$ 
E)  $\arcsin \left( \frac { x + 1 } { \sqrt { 33 } } \right) + C$

Question 18

$\text { Determine whether the improper integral } \int _ { 7 } ^ { 8 } \frac { 1 } { \sqrt { 64 - x ^ { 2 } } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.

Accepted Answer

A)  $\frac { \pi } { 2 } - \arcsin \left( \frac { 7 } { 8 } \right)$ 
B)  $\frac { \pi } { 16 } - \arcsin \left( \frac { 7 } { 8 } \right)$ 
C)  $\frac { \pi } { 2 } - \arcsin \left( \frac { 7 } { 64 } \right)$ 
D)  $\frac { \pi } { 16 } - \arcsin \left( \frac { 7 } { 64 } \right)$ 
E)  diverges 
A)  $\frac { \pi } { 2 } - \arcsin \left( \frac { 7 } { 8 } \right)$ 
B)  $\frac { \pi } { 16 } - \arcsin \left( \frac { 7 } { 8 } \right)$ 
C)  $\frac { \pi } { 2 } - \arcsin \left( \frac { 7 } { 64 } \right)$ 
D)  $\frac { \pi } { 16 } - \arcsin \left( \frac { 7 } { 64 } \right)$ 
E)  diverges

Question 19

Find the indefinite integral. $\int \frac { q ^ { 2 } } { q + 9 } d q$

Accepted Answer

A) $81 \ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 9 q + C$ 
B) $\ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 9 q + C$ 
C) $\ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 10 q + C$ 
D) $81 \ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 10 q + C$ 
E)  $81 \ln | q + 9 | - 10 q + C$ 
A) $81 \ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 9 q + C$ 
B) $\ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 9 q + C$ 
C) $\ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 10 q + C$ 
D) $81 \ln | q + 9 | + \frac { q ^ { 2 } } { 2 } - 10 q + C$ 
E)  $81 \ln | q + 9 | - 10 q + C$

Question 20

Find the indefinite integral. $$\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$$

Accepted Answer

A) $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 25 } + C$ 
B)  $\frac { \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
C)  $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
D)  $\frac { \sqrt { \sin 5 \theta } \left( 3 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
E)  $\frac { 2 \sqrt { \sin 5 \theta } \left( 3 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
A) $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 25 } + C$ 
B)  $\frac { \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
C)  $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
D)  $\frac { \sqrt { \sin 5 \theta } \left( 3 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$ 
E)  $\frac { 2 \sqrt { \sin 5 \theta } \left( 3 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$

$\text { Find the indefinite integral } \int 3 x ^ { 4 } \cos 8 \pi x ^ { 5 } d x \text {. }$

Find the indefinite integral. $\int \frac { 3 m } { m - 9 } d m$

$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$

$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } \frac { x ^ { 5 } } { e ^ { 4 x } } \text { using L'Hopital's Rule if necessary. }$

$\text { Determine whether the improper integral } \int _ { 0 } ^ { 2 } \frac { 3 } { x } d x \text { diverges or converges. Evaluate the }$ integral if it converges.

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \pi / 4 } 3 \tan \theta d \theta \text { diverges or converges. }$

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

$\text { Use partial fractions to find } \int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 3 } + 3 x ^ { 2 } } d x$

Find the indefinite integral by making the substitution $x = 6 \tan \theta$ $\int x \sqrt { 36 + x ^ { 2 } } d x$

$\text { Use integration tables to find } \int \frac { 1 } { 5 x ^ { 2 } + 3 x + 3 } d x \text {. }$

Find the indefinite integral by making the substitution $x = 7 \tan \theta$ $\int \frac { x ^ { 3 } } { \sqrt { 49 + x ^ { 2 } } } d x$

Use substitution and partial fractions to find the indefinite integral. $\int \frac { \sqrt { x } } { x - 25 } d x$

$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$

$\text { A population is growing according to the logistic model } N = \frac { 6000 } { 1 + e ^ { 4.2 - 1.6 ^ { t } } } \text {, where } t$ is the time in days. Find the average population over the interval $[ 0,2 ]$ . Round your answer to one decimal place.

Find the indefinite integral. $\int \sec ^ { 4 } 9 x d x$

$\text { Find the indefinite integral } \int \frac { 1 } { \sqrt { 32 - 2 x - x ^ { 2 } } } d x \text {. }$

$\text { Determine whether the improper integral } \int _ { 7 } ^ { 8 } \frac { 1 } { \sqrt { 64 - x ^ { 2 } } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.

Find the indefinite integral. $\int \frac { q ^ { 2 } } { q + 9 } d q$

Find the indefinite integral. $\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$

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 indefinite integral ∫3x4cos⁡8πx5dx. \text { Find the indefinite integral } \int 3 x ^ { 4 } \cos 8 \pi x ^ { 5 } d x \text {. } Find the indefinite integral ∫3x4cos8πx5dx.

Find the indefinite integral. ∫3mm−9dm\int \frac { 3 m } { m - 9 } d m∫m−93m​dm

Evaluate the integral ∫0π/218−7cos⁡θdθ. \text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. } Evaluate the integral ∫0π/2​8−7cosθ1​dθ.

Find the indefinite integral. ∫cos⁡33xdx\int \cos ^ { 3 } 3 x d x∫cos33xdx

Use partial fractions to find ∫6x2+14x−3x3+3x2dx\text { Use partial fractions to find } \int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 3 } + 3 x ^ { 2 } } d x Use partial fractions to find ∫x3+3x26x2+14x−3​dx

Find the indefinite integral by making the substitution x=6tan⁡θx = 6 \tan \thetax=6tanθ ∫x36+x2dx\int x \sqrt { 36 + x ^ { 2 } } d x∫x36+x2​dx

Use integration tables to find ∫15x2+3x+3dx. \text { Use integration tables to find } \int \frac { 1 } { 5 x ^ { 2 } + 3 x + 3 } d x \text {. } Use integration tables to find ∫5x2+3x+31​dx.

Find the indefinite integral by making the substitution x=7tan⁡θx = 7 \tan \thetax=7tanθ ∫x349+x2dx\int \frac { x ^ { 3 } } { \sqrt { 49 + x ^ { 2 } } } d x∫49+x2​x3​dx

Use substitution and partial fractions to find the indefinite integral. ∫xx−25dx\int \frac { \sqrt { x } } { x - 25 } d x∫x−25x​​dx

Complete the square and find ∫xx2+10x+30dx. \text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. } Complete the square and find ∫x2+10x+30​x​dx.

Find the indefinite integral. ∫sec⁡49xdx\int \sec ^ { 4 } 9 x d x∫sec49xdx

Find the indefinite integral ∫132−2x−x2dx. \text { Find the indefinite integral } \int \frac { 1 } { \sqrt { 32 - 2 x - x ^ { 2 } } } d x \text {. } Find the indefinite integral ∫32−2x−x2​1​dx.

Find the indefinite integral. ∫q2q+9dq\int \frac { q ^ { 2 } } { q + 9 } d q∫q+9q2​dq

Find the indefinite integral. ∫cos⁡35θsin⁡5θdθ\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta∫sin5θ​cos35θ​dθ

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

$\text { Find the indefinite integral } \int 3 x ^ { 4 } \cos 8 \pi x ^ { 5 } d x \text {. }$

Find the indefinite integral. $\int \frac { 3 m } { m - 9 } d m$

$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$

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

$\text { Use partial fractions to find } \int \frac { 6 x ^ { 2 } + 14 x - 3 } { x ^ { 3 } + 3 x ^ { 2 } } d x$

Find the indefinite integral by making the substitution $x = 6 \tan \theta$ $\int x \sqrt { 36 + x ^ { 2 } } d x$

$\text { Use integration tables to find } \int \frac { 1 } { 5 x ^ { 2 } + 3 x + 3 } d x \text {. }$

Find the indefinite integral by making the substitution $x = 7 \tan \theta$ $\int \frac { x ^ { 3 } } { \sqrt { 49 + x ^ { 2 } } } d x$

Use substitution and partial fractions to find the indefinite integral. $\int \frac { \sqrt { x } } { x - 25 } d x$

$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$

Find the indefinite integral. $\int \sec ^ { 4 } 9 x d x$

$\text { Find the indefinite integral } \int \frac { 1 } { \sqrt { 32 - 2 x - x ^ { 2 } } } d x \text {. }$

Find the indefinite integral. $\int \frac { q ^ { 2 } } { q + 9 } d q$

Find the indefinite integral. $\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$