Question 1

Evaluate the limit $\lim _ { x \rightarrow 4 ^ { + } } ( 8 ( x - 4 ) ) ^ { x - 4 }$ using L'Hopital's Rule if necessary.

Accepted Answer

A)  1 
B)  $e$ 
C)  0 
D)  $e ^ { 4 }$ 
E)  $e ^ { 8 }$ 
A)  1 
B)  $e$ 
C)  0 
D)  $e ^ { 4 }$ 
E)  $e ^ { 8 }$

Question 2

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

Accepted Answer

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

Question 3

Find the indefinite integral. 
$$\int s \sqrt { 4 s + 3 } d s$$

Accepted Answer

A)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s - 1 ) } { 20 } + C$ 
B)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s - 2 ) } { 20 } + C$ 
C)  $\frac { ( 4 s - 3 ) ^ { 3 / 2 } ( 2 s + 1 ) } { 20 } + C$ 
D)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s + 2 ) } { 20 } + C$ 
E)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s + 1 ) } { 20 } + C$ 
A)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s - 1 ) } { 20 } + C$ 
B)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s - 2 ) } { 20 } + C$ 
C)  $\frac { ( 4 s - 3 ) ^ { 3 / 2 } ( 2 s + 1 ) } { 20 } + C$ 
D)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s + 2 ) } { 20 } + C$ 
E)  $\frac { ( 4 s + 3 ) ^ { 3 / 2 } ( 2 s + 1 ) } { 20 } + C$

Question 4

Find the indefinite integral. $$\int \tan ^ { 3 } \left( \frac { \pi x } { 2 } \right) \sec ^ { 2 } \left( \frac { \pi x } { 2 } \right) d x$$

Accepted Answer

A)  $- \frac { 1 } { 2 \pi } \tan ^ { 4 } \frac { \pi x } { 2 } + C$ 
B)  $\frac { 1 } { 3 \pi } \tan ^ { 4 } \frac { \pi \pi } { 2 } + C$ 
C)  $\frac { 1 } { 3 \pi } \tan ^ { 5 } \frac { \pi K } { 2 } + C$ 
D)  $\frac { 1 } { 2 \pi } \tan ^ { 4 } \frac { \pi \pi } { 2 } + C$ 
E)  $- \frac { 1 } { 3 \pi } \tan ^ { 4 } \frac { \pi x } { 2 } + C$ 
A)  $- \frac { 1 } { 2 \pi } \tan ^ { 4 } \frac { \pi x } { 2 } + C$ 
B)  $\frac { 1 } { 3 \pi } \tan ^ { 4 } \frac { \pi \pi } { 2 } + C$ 
C)  $\frac { 1 } { 3 \pi } \tan ^ { 5 } \frac { \pi K } { 2 } + C$ 
D)  $\frac { 1 } { 2 \pi } \tan ^ { 4 } \frac { \pi \pi } { 2 } + C$ 
E)  $- \frac { 1 } { 3 \pi } \tan ^ { 4 } \frac { \pi x } { 2 } + C$

Question 5

$\text { Find the indefinite integral by making the substitution } x = 3 \sec \theta \text {. }$ $$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } - 9 } } d x$$

Accepted Answer

A)  $\frac { \sqrt { x ^ { 2 } - 9 } ( x + 9 ) } { 3 } + C$ 
B)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 18 \right) } { 3 } + C$ 
C)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 18 \right) } { 2 } + C$ 
D)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 9 \right) } { 3 } + C$ 
E)  $\frac { \sqrt { x ^ { 2 } - 9 } ( x + 18 ) } { 3 } + C$ 
A)  $\frac { \sqrt { x ^ { 2 } - 9 } ( x + 9 ) } { 3 } + C$ 
B)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 18 \right) } { 3 } + C$ 
C)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 18 \right) } { 2 } + C$ 
D)  $\frac { \sqrt { x ^ { 2 } - 9 } \left( x ^ { 2 } + 9 \right) } { 3 } + C$ 
E)  $\frac { \sqrt { x ^ { 2 } - 9 } ( x + 18 ) } { 3 } + C$

Question 6

$\text { Evaluate } \int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x \text {. }$

Accepted Answer

A)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 ( 1 - \ln 2 )$ 
B) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 - \ln 2$ 
C) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 26 ( 1 + \ln 2 )$ 
D) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 ( 1 + \ln 2 )$ 
E) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 + \ln 2$ 
A)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 ( 1 - \ln 2 )$ 
B) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 - \ln 2$ 
C) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 26 ( 1 + \ln 2 )$ 
D) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 ( 1 + \ln 2 )$ 
E) $\int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x = 13 + \ln 2$

Question 7

$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$

Accepted Answer

A)  $4 \tan \left( t ^ { 7 } \right) + 4 \sec \left( t ^ { 7 } \right) + C$ 
B)  $28 \tan \left( t ^ { 7 } \right) + 28 \sec \left( t ^ { 7 } \right) + C$ 
C)  $4 \tan \left( t ^ { 6 } \right) - 4 \sec \left( t ^ { 6 } \right) + C$ 
D)  $4 \tan \left( t ^ { 7 } \right) \pm 4 \sec \left( t ^ { 7 } \right) + C$ 
E)  $4 \tan \left( t ^ { 7 } \right) - 4 \sec \left( t ^ { 7 } \right) + C$ 
A)  $4 \tan \left( t ^ { 7 } \right) + 4 \sec \left( t ^ { 7 } \right) + C$ 
B)  $28 \tan \left( t ^ { 7 } \right) + 28 \sec \left( t ^ { 7 } \right) + C$ 
C)  $4 \tan \left( t ^ { 6 } \right) - 4 \sec \left( t ^ { 6 } \right) + C$ 
D)  $4 \tan \left( t ^ { 7 } \right) \pm 4 \sec \left( t ^ { 7 } \right) + C$ 
E)  $4 \tan \left( t ^ { 7 } \right) - 4 \sec \left( t ^ { 7 } \right) + C$

Question 8

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

Accepted Answer

A)  $- 4 \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
B)  $4 \left( x ^ { 2 } - 2 x + 2 \right) e ^ { - x } + C$ 
C)  $- 4 x \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
D)  $\left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
E)  $- \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
A)  $- 4 \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
B)  $4 \left( x ^ { 2 } - 2 x + 2 \right) e ^ { - x } + C$ 
C)  $- 4 x \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
D)  $\left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$ 
E)  $- \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$

Question 9

Find the indefinite integral. $$\int \sin ^ { 3 } 19 \theta \sqrt { \cos 19 \theta } d \theta$$

Accepted Answer

A) $- \frac { 1 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } + \frac { 1 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
B) $\frac { \sin ^ { 4 } 19 \theta ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } } { 76 } + C$ 
C)  $\frac { 2 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } - \frac { 2 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
D)  $- \frac { 2 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } + \frac { 2 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
E)  $\frac { 4 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } - \frac { 4 } { 57 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
A) $- \frac { 1 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } + \frac { 1 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
B) $\frac { \sin ^ { 4 } 19 \theta ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } } { 76 } + C$ 
C)  $\frac { 2 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } - \frac { 2 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
D)  $- \frac { 2 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } + \frac { 2 } { 133 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$ 
E)  $\frac { 4 } { 57 } ( \cos 19 \theta ) ^ { \frac { 3 } { 2 } } - \frac { 4 } { 57 } ( \cos 19 \theta ) ^ { \frac { 7 } { 2 } } + C$

Question 10

Find the indefinite integral. $$\int x ^ { 3 } e ^ { 6 x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 36 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$ 
B)  $- \frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( x ^ { 2 } - 6 \right) + C$ 
C)  $\frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( x ^ { 2 } - 6 \right) + C$ 
D)  $\frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$ 
E)  $- \frac { 1 } { 36 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$ 
A)  $\frac { 1 } { 36 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$ 
B)  $- \frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( x ^ { 2 } - 6 \right) + C$ 
C)  $\frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( x ^ { 2 } - 6 \right) + C$ 
D)  $\frac { 1 } { 72 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$ 
E)  $- \frac { 1 } { 36 } e ^ { 6 x ^ { 2 } } \left( 6 x ^ { 2 } - 1 \right) + C$

Question 11

$\text { Find the indefinite integral by making the substitution } x = 7 \sin \theta \text {. }$ $$\int \frac { x } { \left( 49 - x ^ { 2 } \right) ^ { 3 / 2 } } d x$$

Accepted Answer

A)  $\frac { 1 } { \sqrt { 49 - x ^ { 2 } } } + C$ 
B)  $\frac { 3 } { 2 \sqrt { 49 - x ^ { 2 } } } + C$ 
C)  $- \frac { 3 } { 2 \sqrt { 49 - x ^ { 2 } } } + C$ 
D)  $\frac { 4 } { \sqrt { 49 - x ^ { 2 } } } + C$ 
E)  $\frac { 2 } { \sqrt { 49 - x ^ { 2 } } } + C$ 
A)  $\frac { 1 } { \sqrt { 49 - x ^ { 2 } } } + C$ 
B)  $\frac { 3 } { 2 \sqrt { 49 - x ^ { 2 } } } + C$ 
C)  $- \frac { 3 } { 2 \sqrt { 49 - x ^ { 2 } } } + C$ 
D)  $\frac { 4 } { \sqrt { 49 - x ^ { 2 } } } + C$ 
E)  $\frac { 2 } { \sqrt { 49 - x ^ { 2 } } } + C$

Question 12

The predicted cost C (in hundreds of thousands of dollars) for a company to remove p% of a chemical from its waste water is shown in the table below. A model for the data is given by $C = \frac { 128 p } { ( 12 + p ) ( 101 - p ) } , 0 \leq p < 100$. Use the model to find the average cost for removing between
$75 \%$ and $80 \%$ of the chemical.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline p & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \
\hline C & 0 & 0.6 & 1 & 1.3 & 1.6 & 2 & 2.6 & 3.5 & 5.3 & 10.3 \
\hline
\end{array}$

Accepted Answer

A)  $\$ 470,000$ 
B)  $\$ 235,000$ 
C)  $\$ 117,500$ 
D)  $\$ 470,001$ 
E)  $\$ 940,000$ 
A)  $\$ 470,000$ 
B)  $\$ 235,000$ 
C)  $\$ 117,500$ 
D)  $\$ 470,001$ 
E)  $\$ 940,000$

Question 13

Find the integral using integration by parts. $$\int x ^ { 4 } \ln x d x$$

Accepted Answer

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

Question 14

$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } 7 x \sin \left( \frac { 8 } { x } \right) \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $\frac { 8 } { 7 }$ 
B)  56 
C)  $\infty$ 
D)  0 
E)  does not exist 
A)  $\frac { 8 } { 7 }$ 
B)  56 
C)  $\infty$ 
D)  0 
E)  does not exist

Question 15

$\text { Find the area of the region bounded by the graphs of } y = \frac { 16 } { x ^ { 2 } + 5 x + 6 } , y = 0 , x = 0 \text {, }$ and $x = 1$.

Accepted Answer

A)  
$A = 16 \log \left( \frac { 9 } { 8 } \right)$ 
B) 
$A = 16 \log \left( \frac { 8 } { 9 } \right)$ 
C) 
$A = 16 + \log \left( \frac { 5 } { 8 } \right)$ 
D) 
$A = 16 \log \left( \frac { 5 } { 6 } \right)$ 
E) 
$A = 16 - \log \left( \frac { 5 } { 9 } \right)$ 
A)  
$A = 16 \log \left( \frac { 9 } { 8 } \right)$ 
B) 
$A = 16 \log \left( \frac { 8 } { 9 } \right)$ 
C) 
$A = 16 + \log \left( \frac { 5 } { 8 } \right)$ 
D) 
$A = 16 \log \left( \frac { 5 } { 6 } \right)$ 
E) 
$A = 16 - \log \left( \frac { 5 } { 9 } \right)$

Question 16

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

Accepted Answer

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

Question 17

$\text { Suppose the capitalized cost } C \text { is given by } C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - n  } d t \text {, where } C _ { 0 } \text { is the }$ 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 $C _ { 0 } = 670,000 , c ( t ) = 22,000$, and $r = 0.09$. Round your answer to the nearest dollar.

Accepted Answer

A)  $\$ 914,433$ 
B)  $\$ 671,980$ 
C)  $\$ 914,456$ 
D)  $\$ 914,444$ 
E)  $\$ 693,980$ 
A)  $\$ 914,433$ 
B)  $\$ 671,980$ 
C)  $\$ 914,456$ 
D)  $\$ 914,444$ 
E)  $\$ 693,980$

Question 18

$\text { Evaluate the } \lim _ { x \rightarrow \infty } \frac { 2 x - 6 } { 4 x ^ { 2 } + 8 x + 5 } \text { using L'Hôpital's Rule if necessary. }$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\infty$ 
E)  0 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\infty$ 
E)  0

Question 19

Find the indefinite integral. $$\int \frac { \cot \left( 13 / z ^ { 2 } \right) } { z ^ { 3 } } d z$$

Accepted Answer

A)  $\frac { 1 } { 13 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
B)  $- \frac { 1 } { 26 } \ln | \sin ( 13 / z ) | + C$ 
C)  $- \frac { 1 } { 13 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
D)  $- \frac { 1 } { 26 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
E)  $\frac { 1 } { 26 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
A)  $\frac { 1 } { 13 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
B)  $- \frac { 1 } { 26 } \ln | \sin ( 13 / z ) | + C$ 
C)  $- \frac { 1 } { 13 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
D)  $- \frac { 1 } { 26 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$ 
E)  $\frac { 1 } { 26 } \ln \left| \sin \left( 13 / z ^ { 2 } \right) \right| + C$

Question 20

$\text { Find the indefinite integral by making the substitution } x = 9 \sin \theta \text {. }$ $$\int \frac { \sqrt { 81 - x ^ { 2 } } } { x } d x$$

Accepted Answer

A)  $\sqrt{81-x^{2}}-9 \ln \left|\frac{9-\sqrt{81-x^{2}}}{x}\right|+C$ 
B) $\sqrt { 81 - x ^ { 2 } } - 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$
$\left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } + 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
C) $\sqrt { 81 - x ^ { 2 } } + 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
D) $\sqrt { 81 - x ^ { 2 } } + 9 \ln \left| \frac { 9 + \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
E) $\left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } - 9 \ln \left| \frac { 9 + \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
A)  $\sqrt{81-x^{2}}-9 \ln \left|\frac{9-\sqrt{81-x^{2}}}{x}\right|+C$ 
B) $\sqrt { 81 - x ^ { 2 } } - 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$
$\left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } + 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
C) $\sqrt { 81 - x ^ { 2 } } + 9 \ln \left| \frac { 9 - \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
D) $\sqrt { 81 - x ^ { 2 } } + 9 \ln \left| \frac { 9 + \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$ 
E) $\left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } - 9 \ln \left| \frac { 9 + \sqrt { 81 - x ^ { 2 } } } { x } \right| + C$

Evaluate the limit $\lim _ { x \rightarrow 4 ^ { + } } ( 8 ( x - 4 ) ) ^ { x - 4 }$ using L'Hopital's Rule if necessary.

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

Find the indefinite integral. $\int s \sqrt { 4 s + 3 } d s$

Find the indefinite integral. $\int \tan ^ { 3 } \left( \frac { \pi x } { 2 } \right) \sec ^ { 2 } \left( \frac { \pi x } { 2 } \right) d x$

$\text { Find the indefinite integral by making the substitution } x = 3 \sec \theta \text {. }$ $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } - 9 } } d x$

$\text { Evaluate } \int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x \text {. }$

$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$

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

Find the indefinite integral. $\int \sin ^ { 3 } 19 \theta \sqrt { \cos 19 \theta } d \theta$

Find the indefinite integral. $\int x ^ { 3 } e ^ { 6 x ^ { 2 } } d x$

$\text { Find the indefinite integral by making the substitution } x = 7 \sin \theta \text {. }$ $\int \frac { x } { \left( 49 - x ^ { 2 } \right) ^ { 3 / 2 } } d x$

Find the integral using integration by parts. $\int x ^ { 4 } \ln x d x$

$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } 7 x \sin \left( \frac { 8 } { x } \right) \text { using L'Hopital's Rule if necessary. }$

$\text { Find the area of the region bounded by the graphs of } y = \frac { 16 } { x ^ { 2 } + 5 x + 6 } , y = 0 , x = 0 \text {, }$ and $x = 1$ .

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

$\text { Evaluate the } \lim _ { x \rightarrow \infty } \frac { 2 x - 6 } { 4 x ^ { 2 } + 8 x + 5 } \text { using L'Hôpital's Rule if necessary. }$

Find the indefinite integral. $\int \frac { \cot \left( 13 / z ^ { 2 } \right) } { z ^ { 3 } } d z$

$\text { Find the indefinite integral by making the substitution } x = 9 \sin \theta \text {. }$ $\int \frac { \sqrt { 81 - x ^ { 2 } } } { x } d x$

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

Evaluate the limit lim⁡x→4+(8(x−4))x−4\lim _ { x \rightarrow 4 ^ { + } } ( 8 ( x - 4 ) ) ^ { x - 4 }limx→4+​(8(x−4))x−4 using L'Hopital's Rule if necessary.

Find the indefinite integral. ∫cos⁡xsin⁡3xdx\int \cos x \sin ^ { 3 } x d x∫cosxsin3xdx

Find the indefinite integral. ∫s4s+3ds\int s \sqrt { 4 s + 3 } d s∫s4s+3​ds

Find the indefinite integral. ∫tan⁡3(πx2)sec⁡2(πx2)dx\int \tan ^ { 3 } \left( \frac { \pi x } { 2 } \right) \sec ^ { 2 } \left( \frac { \pi x } { 2 } \right) d x∫tan3(2πx​)sec2(2πx​)dx

Evaluate ∫0π426tan⁡3xdx. \text { Evaluate } \int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x \text {. } Evaluate ∫04π​​26tan3xdx.

Use integration tables to find ∫28t61+sin⁡(t7)dt. \text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. } Use integration tables to find ∫1+sin(t7)28t6​dt.

Find the indefinite integral. ∫4x2exdx\int \frac { 4 x ^ { 2 } } { e ^ { x } } d x∫ex4x2​dx

Find the indefinite integral. ∫sin⁡319θcos⁡19θdθ\int \sin ^ { 3 } 19 \theta \sqrt { \cos 19 \theta } d \theta∫sin319θcos19θ​dθ

Find the indefinite integral. ∫x3e6x2dx\int x ^ { 3 } e ^ { 6 x ^ { 2 } } d x∫x3e6x2dx

Find the integral using integration by parts. ∫x4ln⁡xdx\int x ^ { 4 } \ln x d x∫x4lnxdx

Find the area of the region bounded by the graphs of y=16x2+5x+6,y=0,x=0, \text { Find the area of the region bounded by the graphs of } y = \frac { 16 } { x ^ { 2 } + 5 x + 6 } , y = 0 , x = 0 \text {, } Find the area of the region bounded by the graphs of y=x2+5x+616​,y=0,x=0, and x=1x = 1x=1 .

Find the indefinite integral. ∫sin⁡23xdx\int \sin ^ { 2 } 3 x d x∫sin23xdx

Find the indefinite integral. ∫cot⁡(13/z2)z3dz\int \frac { \cot \left( 13 / z ^ { 2 } \right) } { z ^ { 3 } } d z∫z3cot(13/z2)​dz

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

Evaluate the limit $\lim _ { x \rightarrow 4 ^ { + } } ( 8 ( x - 4 ) ) ^ { x - 4 }$ using L'Hopital's Rule if necessary.

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

Find the indefinite integral. $\int s \sqrt { 4 s + 3 } d s$

Find the indefinite integral. $\int \tan ^ { 3 } \left( \frac { \pi x } { 2 } \right) \sec ^ { 2 } \left( \frac { \pi x } { 2 } \right) d x$

$\text { Evaluate } \int _ { 0 } ^ { \frac { \pi } { 4 } } 26 \tan ^ { 3 } x d x \text {. }$

$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$

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

Find the indefinite integral. $\int \sin ^ { 3 } 19 \theta \sqrt { \cos 19 \theta } d \theta$

Find the indefinite integral. $\int x ^ { 3 } e ^ { 6 x ^ { 2 } } d x$

Find the integral using integration by parts. $\int x ^ { 4 } \ln x d x$

$\text { Find the area of the region bounded by the graphs of } y = \frac { 16 } { x ^ { 2 } + 5 x + 6 } , y = 0 , x = 0 \text {, }$ and $x = 1$ .

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

Find the indefinite integral. $\int \frac { \cot \left( 13 / z ^ { 2 } \right) } { z ^ { 3 } } d z$