Question 1

Find the indefinite integral. $\int \frac { x ^ { 2 } } { x - 5 } d x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 2 } + 5 x + ( x + 1 ) \ln ( x - 5 ) + C$ 
B)  $\frac { x ^ { 2 } } { 3 } + x + 25 \ln ( x - 5 ) + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - x - 25 \ln ( x + 5 ) + C$ 
D)  $\frac { x ^ { 2 } } { 3 } + 5 x - 25 \ln ( x - 5 ) + C$ 
E) $\frac { x ^ { 2 } } { 2 } + 5 x + 25 \ln ( x - 5 ) + C$ 
A)  $\frac { x ^ { 2 } } { 2 } + 5 x + ( x + 1 ) \ln ( x - 5 ) + C$ 
B)  $\frac { x ^ { 2 } } { 3 } + x + 25 \ln ( x - 5 ) + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - x - 25 \ln ( x + 5 ) + C$ 
D)  $\frac { x ^ { 2 } } { 3 } + 5 x - 25 \ln ( x - 5 ) + C$ 
E) $\frac { x ^ { 2 } } { 2 } + 5 x + 25 \ln ( x - 5 ) + C$

Question 2

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.

Accepted Answer

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

Question 3

A hydraulic cylinder on an industrial machine pushes a steel block a distance of x feet $( 0 \leq x \leq 4 )$, where the variable force required is $F ( x ) = 3000 x e ^ { - x }$ pounds. Find the work done in pushing the block the full 4 feet through the machine. Round your answer to three decimal places.

Accepted Answer

A)  $2725.265 \mathrm { ft } . \mathrm { lbs }$ 
B)  $2890.106 \mathrm { ft } . \mathrm { lbs }$ 
C)  $1655.749 \mathrm { ft } . \mathrm { lbs }$ 
D)  $2839.235 \mathrm { ft } . \mathrm { lbs }$ 
E)  $3219.788 \mathrm { ft }$.lbs 
A)  $2725.265 \mathrm { ft } . \mathrm { lbs }$ 
B)  $2890.106 \mathrm { ft } . \mathrm { lbs }$ 
C)  $1655.749 \mathrm { ft } . \mathrm { lbs }$ 
D)  $2839.235 \mathrm { ft } . \mathrm { lbs }$ 
E)  $3219.788 \mathrm { ft }$.lbs

Question 4

Determine whether the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$ diverges or converges. Evaluate the integral if it converges.

Accepted Answer

A)  9 
B)  $e ^ { - 1 / 3 }$ 
C)  $- \frac { 1 } { 3 }$ 
D)  3 
E)  diverges 
A)  9 
B)  $e ^ { - 1 / 3 }$ 
C)  $- \frac { 1 } { 3 }$ 
D)  3 
E)  diverges

Question 5

$\text { Evaluate the limit } \lim _ { x \rightarrow - \infty } \frac { 2 e ^ { - \frac { 1 } { 2 } x } } { x ^ { 3 } } \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 24 }$ 
C)  $- \infty$ 
D)  $\infty$ 
E)  2 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 24 }$ 
C)  $- \infty$ 
D)  $\infty$ 
E)  2

Question 6

Determine whether the improper integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$ diverges or converges.

Accepted Answer

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

Question 7

$\text { Use integration tables to find } \int ( 2 x + 5 ) ^ { 2 } \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } d x \text {. }$

Accepted Answer

A)  $\frac { 1 } { 32 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } + 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } \right| \right] + C$ 
B)  $\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } - 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } \right| \right] + C$ 
C)  $\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } \pm 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } \pm 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } \pm 9 } \right| \right] + C$ 
D) 
$\frac { 1 } { 8 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } - 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } \right| \right] + C$ 
E) 
$\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } + 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } \right| \right] + C$ 
A)  $\frac { 1 } { 32 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } + 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } \right| \right] + C$ 
B)  $\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } - 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } \right| \right] + C$ 
C)  $\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } \pm 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } \pm 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } \pm 9 } \right| \right] + C$ 
D) 
$\frac { 1 } { 8 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } - 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } \right| \right] + C$ 
E) 
$\frac { 1 } { 16 } \left[ ( 2 x + 5 ) \left( 2 ( 2 x + 5 ) ^ { 2 } + 9 \right) \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } - 81 \ln \left| ( 2 x + 5 ) + \sqrt { ( 2 x + 5 ) ^ { 2 } + 9 } \right| \right] + C$

Question 8

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { 49 - x ^ { 2 } } - 7 } { 9 x } \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $- \frac { 1 } { 9 }$ 
B)  $- \frac { 7 } { 9 }$ 
C)  1 
D)  0 
E)  does not exist 
A)  $- \frac { 1 } { 9 }$ 
B)  $- \frac { 7 } { 9 }$ 
C)  1 
D)  0 
E)  does not exist

Question 9

$\text { Suppose the capitalized cost } C \text { is given by } C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - n  } d t \text {, where } C _ { 0 } \text { is the }$ original investment, $t$ is the time in years, $r$ is the annual interest rate compounded continuously, and $c ( t )$ is the annual cost of maintenance. Find the capitalized cost $C$ of an asset forever if
$C _ { 0 } = 625,000 , c ( t ) = 23,500 ( 1 + 0.07 t )$, and $r = 0.05$. Round your answer to the nearest dollar.

Accepted Answer

A)  681,400 
B)  $1,753,400$ 
C)  $1,799,511$ 
D)  681,420 
E)  $1,753,000$ 
A)  681,400 
B)  $1,753,400$ 
C)  $1,799,511$ 
D)  681,420 
E)  $1,753,000$

Question 10

Find the indefinite integral. $$\int \cos ^ { 3 } 2 x \sin ^ { 2 } 2 x d x$$

Accepted Answer

A)  $\frac { \sin 2 x } { 30 } \left( 4 - 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$ 
B)  $\frac { \sin 2 x } { 10 } \left( 5 \sin ^ { 2 } 2 x - 3 \sin ^ { 4 } 2 x \right) + C$ 
C)  $\frac { \sin 2 x } { 30 } \left( 5 \sin ^ { 2 } 2 x - 3 \sin ^ { 4 } 2 x \right) + C$ 
D)  $\frac { \sin 2 x } { 10 } \left( 4 - 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$ 
E)  $\frac { \sin 2 x } { 30 } \left( 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$ 
A)  $\frac { \sin 2 x } { 30 } \left( 4 - 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$ 
B)  $\frac { \sin 2 x } { 10 } \left( 5 \sin ^ { 2 } 2 x - 3 \sin ^ { 4 } 2 x \right) + C$ 
C)  $\frac { \sin 2 x } { 30 } \left( 5 \sin ^ { 2 } 2 x - 3 \sin ^ { 4 } 2 x \right) + C$ 
D)  $\frac { \sin 2 x } { 10 } \left( 4 - 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$ 
E)  $\frac { \sin 2 x } { 30 } \left( 5 \sin ^ { 2 } 2 x + 3 \sin ^ { 4 } 2 x \right) + C$

Question 11

Find the indefinite integral. $$\int \sin ^ { 3 } \frac { x } { 8 } d x$$

Accepted Answer

A)  $\frac { 8 \left( 1 + \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
B)  $- \frac { 8 \left( 3 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
C)  $\frac { 8 \left( 2 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
D)  $\frac { 8 \left( 3 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
E) $- \frac { 8 \left( 2 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
A)  $\frac { 8 \left( 1 + \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
B)  $- \frac { 8 \left( 3 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
C)  $\frac { 8 \left( 2 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
D)  $\frac { 8 \left( 3 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$ 
E) $- \frac { 8 \left( 2 - \cos ^ { 2 } \frac { x } { 8 } \right) \cos \frac { x } { 8 } } { 3 } + C$

Question 12

$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$

Accepted Answer

A)  $\frac { 1 } { 8 } \left( \ln \left| 4 + 5 \sin ( x ) + 4 \sin ^ { 2 } ( x ) \right| - \frac { 10 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) \right) + C$ 
B)  $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \cos ( x ) + 5 } { \sqrt { 39 } } \right) + C$ 
C)  $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \sin ( x ) + 5 - \sqrt { 39 } } { 8 \sin ( x ) + 5 + \sqrt { 39 } } \right| + C$ 
D)  $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) + C$ 
E) $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \cos ( x ) + 5 - \sqrt { 39 } } { 8 \cos ( x ) + 5 + \sqrt { 39 } } \right| + C$ 
A)  $\frac { 1 } { 8 } \left( \ln \left| 4 + 5 \sin ( x ) + 4 \sin ^ { 2 } ( x ) \right| - \frac { 10 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) \right) + C$ 
B)  $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \cos ( x ) + 5 } { \sqrt { 39 } } \right) + C$ 
C)  $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \sin ( x ) + 5 - \sqrt { 39 } } { 8 \sin ( x ) + 5 + \sqrt { 39 } } \right| + C$ 
D)  $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) + C$ 
E) $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \cos ( x ) + 5 - \sqrt { 39 } } { 8 \cos ( x ) + 5 + \sqrt { 39 } } \right| + C$

Question 13

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } e ^ { - 3 x } \sin ( 3 x ) d x \text { diverges or converges. }$ Evaluate the integral if it converges.

Accepted Answer

A)  18 
B)  9 
C)  $\frac { 1 } { 6 }$ 
D)  3 
E)  diverges 
A)  18 
B)  9 
C)  $\frac { 1 } { 6 }$ 
D)  3 
E)  diverges

Question 14

$\text { Solve the differential equation } \frac { d y } { d x } = \left( e ^ { x } + 8 \right) ^ { 2 } \text {. }$

Accepted Answer

A)  $y = e ^ { 2 x } + 8 e ^ { x } + 64 x + C$ 
B)  $y = \frac { 1 } { 2 } e ^ { 2 x } + 16 e ^ { x } + 64 x + C$ 
C)  $y = e ^ { 2 x } + 64 e ^ { x } + 16 x + C$ 
D)  $y = \frac { 1 } { 2 } x e ^ { 2 x } + 16 e ^ { x } + 8 x + C$ 
E)  $y = \frac { 1 } { 2 } x e ^ { 2 x } + 8 e ^ { x } + 64 x + C$ 
A)  $y = e ^ { 2 x } + 8 e ^ { x } + 64 x + C$ 
B)  $y = \frac { 1 } { 2 } e ^ { 2 x } + 16 e ^ { x } + 64 x + C$ 
C)  $y = e ^ { 2 x } + 64 e ^ { x } + 16 x + C$ 
D)  $y = \frac { 1 } { 2 } x e ^ { 2 x } + 16 e ^ { x } + 8 x + C$ 
E)  $y = \frac { 1 } { 2 } x e ^ { 2 x } + 8 e ^ { x } + 64 x + C$

Question 15

$\text { Evaluate the limit } \lim _ { x \rightarrow 8 ^ { + } } \left( \frac { 21 x - 40 } { x ^ { 2 } - 64 } - \frac { x } { x - 8 } \right) \text {, using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $- \frac { 3 } { 8 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $- \frac { 3 } { 16 }$ 
E)  does not exist 
A)  $\frac { 3 } { 16 }$ 
B)  $- \frac { 3 } { 8 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $- \frac { 3 } { 16 }$ 
E)  does not exist

Question 16

Find the indefinite integral. $$\int \frac { w } { \sqrt { 1 + 3 w } } d w$$

Accepted Answer

A) $\frac { ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 9 } + C$ 
B) $\frac { 2 ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 9 } + C$ 
C)  $\frac { 2 ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 27 } + C$ 
D)  $\frac { ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 18 } + C$ 
E)  $\frac { 2 ( 2 - 3 w ) \sqrt { 3 w + 1 } } { 27 } + C$ 
A) $\frac { ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 9 } + C$ 
B) $\frac { 2 ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 9 } + C$ 
C)  $\frac { 2 ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 27 } + C$ 
D)  $\frac { ( 3 w - 2 ) \sqrt { 3 w + 1 } } { 18 } + C$ 
E)  $\frac { 2 ( 2 - 3 w ) \sqrt { 3 w + 1 } } { 27 } + C$

Question 17

Evaluate the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( x ^ { 3 } \right) } { x ^ { 10 } }$ using L'Hopital's Rule if necessary.

Accepted Answer

A)  $\frac { 10 } { 3 }$ 
B)  $\infty$ 
C)  $- \infty$ 
D)  $\frac { 3 } { 10 }$ 
E)  0 
A)  $\frac { 10 } { 3 }$ 
B)  $\infty$ 
C)  $- \infty$ 
D)  $\frac { 3 } { 10 }$ 
E)  0

Question 18

Find the indefinite integral. $$\int x \ln ( x - 8 ) d x$$

Accepted Answer

A)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } + 16 x } { 4 } + C$ 
B)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) + \frac { x ^ { 2 } + 8 x } { 4 } + C$ 
C)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) + \frac { x ^ { 2 } + 16 x } { 4 } + C$ 
D)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } + 16 x } { 2 } + C$ 
E)  $\left( \frac { x ^ { 2 } + 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } - 8 x } { 4 } + C$ 
A)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } + 16 x } { 4 } + C$ 
B)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) + \frac { x ^ { 2 } + 8 x } { 4 } + C$ 
C)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) + \frac { x ^ { 2 } + 16 x } { 4 } + C$ 
D)  $\left( \frac { x ^ { 2 } - 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } + 16 x } { 2 } + C$ 
E)  $\left( \frac { x ^ { 2 } + 64 } { 2 } \right) \ln ( x - 8 ) - \frac { x ^ { 2 } - 8 x } { 4 } + C$

Question 19

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

Accepted Answer

A)  $\frac { x ^ { 5 } } { 25 } [ 4 \ln ( x ) - 1 ] + C$ 
B)  $\frac { x ^ { 4 } } { 25 } \left[ \ln \left( x ^ { 4 } \right) - 1 \right] + C$ 
C)  $\frac { x ^ { 5 } } { 25 } [ 5 \ln ( x ) - 1 ] + C$ 
D)  $\frac { x ^ { 5 } } { 16 } \left[ \ln \left( x ^ { 5 } \right) - 1 \right] + C$ 
E)  $\frac { x ^ { 3 } } { 25 } \left[ \ln \left( x ^ { 3 } \right) - 1 \right] + C$ 
A)  $\frac { x ^ { 5 } } { 25 } [ 4 \ln ( x ) - 1 ] + C$ 
B)  $\frac { x ^ { 4 } } { 25 } \left[ \ln \left( x ^ { 4 } \right) - 1 \right] + C$ 
C)  $\frac { x ^ { 5 } } { 25 } [ 5 \ln ( x ) - 1 ] + C$ 
D)  $\frac { x ^ { 5 } } { 16 } \left[ \ln \left( x ^ { 5 } \right) - 1 \right] + C$ 
E)  $\frac { x ^ { 3 } } { 25 } \left[ \ln \left( x ^ { 3 } \right) - 1 \right] + C$

Question 20

Solve. $$y ^ { t } = \tan ^ { 3 } 3 x \sec 3 x$$

Accepted Answer

A)  $y = \frac { 1 } { 6 } \sec ^ { 3 } 3 x - \frac { 1 } { 4 } \sec 3 x + C$ 
B)  $y = \frac { 1 } { 9 } \sec ^ { 2 } 3 x - \frac { 1 } { 4 } \sec 3 x + C$ 
C)  $y = \frac { 1 } { 6 } \sec ^ { 2 } 3 x + \frac { 1 } { 3 } \sec 3 x + C$ 
D)  $y = \frac { 1 } { 9 } \sec ^ { 3 } 3 x - \frac { 1 } { 3 } \sec 3 x + C$ 
E)  $y = \frac { 1 } { 9 } \sec ^ { 3 } 3 x + \frac { 1 } { 3 } \sec 3 x + C$ 
A)  $y = \frac { 1 } { 6 } \sec ^ { 3 } 3 x - \frac { 1 } { 4 } \sec 3 x + C$ 
B)  $y = \frac { 1 } { 9 } \sec ^ { 2 } 3 x - \frac { 1 } { 4 } \sec 3 x + C$ 
C)  $y = \frac { 1 } { 6 } \sec ^ { 2 } 3 x + \frac { 1 } { 3 } \sec 3 x + C$ 
D)  $y = \frac { 1 } { 9 } \sec ^ { 3 } 3 x - \frac { 1 } { 3 } \sec 3 x + C$ 
E)  $y = \frac { 1 } { 9 } \sec ^ { 3 } 3 x + \frac { 1 } { 3 } \sec 3 x + C$

Find the indefinite integral. $\int \frac { x ^ { 2 } } { x - 5 } d x$

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.

Determine whether the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$ diverges or converges. Evaluate the integral if it converges.

$\text { Evaluate the limit } \lim _ { x \rightarrow - \infty } \frac { 2 e ^ { - \frac { 1 } { 2 } x } } { x ^ { 3 } } \text { using L'Hopital's Rule if necessary. }$

Determine whether the improper integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$ diverges or converges.

$\text { Use integration tables to find } \int ( 2 x + 5 ) ^ { 2 } \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } d x \text {. }$

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { 49 - x ^ { 2 } } - 7 } { 9 x } \text { using L'Hopital's Rule if necessary. }$

Find the indefinite integral. $\int \cos ^ { 3 } 2 x \sin ^ { 2 } 2 x d x$

Find the indefinite integral. $\int \sin ^ { 3 } \frac { x } { 8 } d x$

$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } e ^ { - 3 x } \sin ( 3 x ) d x \text { diverges or converges. }$ Evaluate the integral if it converges.

$\text { Solve the differential equation } \frac { d y } { d x } = \left( e ^ { x } + 8 \right) ^ { 2 } \text {. }$

$\text { Evaluate the limit } \lim _ { x \rightarrow 8 ^ { + } } \left( \frac { 21 x - 40 } { x ^ { 2 } - 64 } - \frac { x } { x - 8 } \right) \text {, using L'Hopital's Rule if necessary. }$

Find the indefinite integral. $\int \frac { w } { \sqrt { 1 + 3 w } } d w$

Evaluate the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( x ^ { 3 } \right) } { x ^ { 10 } }$ using L'Hopital's Rule if necessary.

Find the indefinite integral. $\int x \ln ( x - 8 ) d x$

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

Solve. $y ^ { t } = \tan ^ { 3 } 3 x \sec 3 x$

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. ∫x2x−5dx\int \frac { x ^ { 2 } } { x - 5 } d x∫x−5x2​dx

Determine whether the improper integral ∫0∞xe−x/3dx\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x∫0∞​xe−x/3dx diverges or converges. Evaluate the integral if it converges.

Determine whether the improper integral ∫013xdx\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x∫01​x3​dx diverges or converges.

Use integration tables to find ∫(2x+5)2(2x+5)2−9dx. \text { Use integration tables to find } \int ( 2 x + 5 ) ^ { 2 } \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } d x \text {. } Use integration tables to find ∫(2x+5)2(2x+5)2−9​dx.

Find the indefinite integral. ∫cos⁡32xsin⁡22xdx\int \cos ^ { 3 } 2 x \sin ^ { 2 } 2 x d x∫cos32xsin22xdx

Find the indefinite integral. ∫sin⁡3x8dx\int \sin ^ { 3 } \frac { x } { 8 } d x∫sin38x​dx

Use integration tables to find ∫cos⁡(x)4sin⁡2(x)+5sin⁡(x)+4dx\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x Use integration tables to find ∫4sin2(x)+5sin(x)+4cos(x)​dx

Solve the differential equation dydx=(ex+8)2. \text { Solve the differential equation } \frac { d y } { d x } = \left( e ^ { x } + 8 \right) ^ { 2 } \text {. } Solve the differential equation dxdy​=(ex+8)2.

Find the indefinite integral. ∫w1+3wdw\int \frac { w } { \sqrt { 1 + 3 w } } d w∫1+3w​w​dw

Evaluate the limit lim⁡x→∞ln⁡(x3)x10\lim _ { x \rightarrow \infty } \frac { \ln \left( x ^ { 3 } \right) } { x ^ { 10 } }limx→∞​x10ln(x3)​ using L'Hopital's Rule if necessary.

Find the indefinite integral. ∫xln⁡(x−8)dx\int x \ln ( x - 8 ) d x∫xln(x−8)dx

Find the indefinite integral. ∫x4ln⁡xdx\int x ^ { 4 } \ln x d x∫x4lnxdx

Solve. yt=tan⁡33xsec⁡3xy ^ { t } = \tan ^ { 3 } 3 x \sec 3 xyt=tan33xsec3x

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

Find the indefinite integral. $\int \frac { x ^ { 2 } } { x - 5 } d x$

Determine whether the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$ diverges or converges. Evaluate the integral if it converges.

Determine whether the improper integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$ diverges or converges.

$\text { Use integration tables to find } \int ( 2 x + 5 ) ^ { 2 } \sqrt { ( 2 x + 5 ) ^ { 2 } - 9 } d x \text {. }$

Find the indefinite integral. $\int \cos ^ { 3 } 2 x \sin ^ { 2 } 2 x d x$

Find the indefinite integral. $\int \sin ^ { 3 } \frac { x } { 8 } d x$

$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$

$\text { Solve the differential equation } \frac { d y } { d x } = \left( e ^ { x } + 8 \right) ^ { 2 } \text {. }$

Find the indefinite integral. $\int \frac { w } { \sqrt { 1 + 3 w } } d w$

Evaluate the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( x ^ { 3 } \right) } { x ^ { 10 } }$ using L'Hopital's Rule if necessary.

Find the indefinite integral. $\int x \ln ( x - 8 ) d x$

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

Solve. $y ^ { t } = \tan ^ { 3 } 3 x \sec 3 x$