Question 1

Find the following indefinite integral. $$\int x ^ { 7 } \left( 4 ^ { - x ^ { 8 } } \right) d x$$

Accepted Answer

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

Question 2

Evaluate the definite integral. $\int _ { 1 } ^ { 2 } \frac { 1 - \cos ( 9 \theta ) } { 9 \theta - \sin ( 9 \theta ) } d \theta$

Accepted Answer

A)  $\frac { 1 } { 9 } \ln \left| \frac { 1 - \cos ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
B)  $\frac { 1 } { 9 } \ln \left| \frac { 1 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
C)  $\frac { 1 } { 18 } \ln \left| \frac { 18 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
D)  $\frac { 1 } { 9 } \ln \left| \frac { 18 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
E)  $\frac { 1 } { 9 } \ln \left| \frac { 2 - \cos ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
A)  $\frac { 1 } { 9 } \ln \left| \frac { 1 - \cos ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
B)  $\frac { 1 } { 9 } \ln \left| \frac { 1 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
C)  $\frac { 1 } { 18 } \ln \left| \frac { 18 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
D)  $\frac { 1 } { 9 } \ln \left| \frac { 18 - \sin ( 18 ) } { 9 - \sin ( 9 ) } \right|$ 
E)  $\frac { 1 } { 9 } \ln \left| \frac { 2 - \cos ( 18 ) } { 9 - \sin ( 9 ) } \right|$

Question 3

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

Accepted Answer

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

Question 4

Solve the differential equation. $\frac { d f } { d z } = \frac { 4 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) ^ { 2 } }$

Accepted Answer

A) $f ( z ) = \frac { 2 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
B) $f ( z ) = \frac { 2 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
C) $f ( z ) = \frac { 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
D) $f ( z ) = \frac { - 1 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
E) $f ( z ) = \frac { 1 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
A) $f ( z ) = \frac { 2 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
B) $f ( z ) = \frac { 2 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
C) $f ( z ) = \frac { 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
D) $f ( z ) = \frac { - 1 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$ 
E) $f ( z ) = \frac { 1 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) } + C$

Question 5

Find the sum given below. $$\sum _ { k = 3 } ^ { 5 } \frac { 5 } { k ^ { 2 } }$$

Accepted Answer

A)  $\frac { 3845 } { 144 }$ 
B)  $\frac { 769 } { 3600 }$ 
C)  $\frac { 769 } { 720 }$ 
D)  $\frac { 553 } { 720 }$ 
E)  $\frac { 769 } { 80 }$ 
A)  $\frac { 3845 } { 144 }$ 
B)  $\frac { 769 } { 3600 }$ 
C)  $\frac { 769 } { 720 }$ 
D)  $\frac { 553 } { 720 }$ 
E)  $\frac { 769 } { 80 }$

Question 6

The graph of the function $g ( y ) = y ^ { 3 }$ is given below. Which of the following definite integrals yields the area of the shaded region?

Accepted Answer

A)  $\int _ { - 2 } ^ { 2 } y ^ { 3 } d y$ 
B)  $\int _ { - 2 } ^ { 0 } y ^ { 3 } d y$ 
C)  $\int _ { 0 } ^ { 8 } y ^ { 3 } d y$ 
D)  $\int _ { 0 } ^ { 2 } y ^ { 3 } d x$ 
E)  $\int _ { 0 } ^ { 2 } y ^ { 3 } d y$ 
A)  $\int _ { - 2 } ^ { 2 } y ^ { 3 } d y$ 
B)  $\int _ { - 2 } ^ { 0 } y ^ { 3 } d y$ 
C)  $\int _ { 0 } ^ { 8 } y ^ { 3 } d y$ 
D)  $\int _ { 0 } ^ { 2 } y ^ { 3 } d x$ 
E)  $\int _ { 0 } ^ { 2 } y ^ { 3 } d y$

Question 7

Evaluate the definite integral 
$\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$. Use a graphing utility to verify your results.

Accepted Answer

A) $\frac { 40 } { 1,323 }$ 
B)  $\frac { 160 } { 1,323 }$ 
C)  $\frac { 20 } { 441 }$ 
D)  $\frac { 80 } { 441 }$ 
E)  $\frac { 160 } { 441 }$ 
A) $\frac { 40 } { 1,323 }$ 
B)  $\frac { 160 } { 1,323 }$ 
C)  $\frac { 20 } { 441 }$ 
D)  $\frac { 80 } { 441 }$ 
E)  $\frac { 160 } { 441 }$

Question 8

Find the indefinite integral $$\int \sqrt [ 13 ] { x ^ { 8 } } d x$$

Accepted Answer

A)  $\frac { 8 } { 13 } x ^ { 21 } + C$ 
B)  $\frac { 21 } { 13 } x ^ { 13 / 21 } + C$ 
C)  $\frac { 8 } { 21 } x ^ { 8 / 21 } + C$ 
D)  $\frac { 13 } { 8 } x ^ { 21 } + C$ 
E)  $\frac { 13 } { 21 } x ^ { 21 / 13 } + C$ 
A)  $\frac { 8 } { 13 } x ^ { 21 } + C$ 
B)  $\frac { 21 } { 13 } x ^ { 13 / 21 } + C$ 
C)  $\frac { 8 } { 21 } x ^ { 8 / 21 } + C$ 
D)  $\frac { 13 } { 8 } x ^ { 21 } + C$ 
E)  $\frac { 13 } { 21 } x ^ { 21 / 13 } + C$

Question 9

Solve the differential equation. $$\frac { d y } { d t } = - 12 t ^ { 2 } , \quad y ( 0 ) = 5$$

Accepted Answer

A)  $y ( t ) = - 4 t ^ { 3 } - 5$ 
B)  $y ( t ) = - 4 t ^ { 3 } + 5$ 
C)  $y ( t ) = - 4 t ^ { 3 } + 1$ 
D)  $y ( t ) = - 4 t ^ { 3 } + C$ 
E)  $y ( t ) = - 24 t + 5$ 
A)  $y ( t ) = - 4 t ^ { 3 } - 5$ 
B)  $y ( t ) = - 4 t ^ { 3 } + 5$ 
C)  $y ( t ) = - 4 t ^ { 3 } + 1$ 
D)  $y ( t ) = - 4 t ^ { 3 } + C$ 
E)  $y ( t ) = - 24 t + 5$

Question 10

The Force$\ { F }$ (in newtons) of a hydraulic cylinder in a press is proportional to the square of $\sec x$ where $x$ is the distance (in meters) that the cylinder is extended in its cycle. The domain of $F$ is $\left[ 0 , \frac { \pi } { 3 } \right]$ and $F ( 0 ) = 150$. Find $F$ as a function of $x$.

Accepted Answer

A)  $F ( x ) = 150 \sec ^ { 2 } x$ 
B)  $F ( x ) = 150 + \sec ^ { 2 } x$ 
C)  $F ( x ) = 300 \sec ^ { 2 } x$ 
D)  $F ( x ) = \sec ^ { 2 } x + 1$ 
E)  $F ( x ) = 75 \sec ^ { 2 } x$ 
A)  $F ( x ) = 150 \sec ^ { 2 } x$ 
B)  $F ( x ) = 150 + \sec ^ { 2 } x$ 
C)  $F ( x ) = 300 \sec ^ { 2 } x$ 
D)  $F ( x ) = \sec ^ { 2 } x + 1$ 
E)  $F ( x ) = 75 \sec ^ { 2 } x$

Question 11

Evaluate the definite integral. $$\int _ { 1 } ^ { 7 } \frac { e ^ { 7 \sqrt { x } } } { \sqrt { x } } d x$$

Accepted Answer

A)  $e ^ { 7 \sqrt { 7 } } - e ^ { 7 }$ 
B)  $\frac { e ^ { 7 \sqrt { 7 } } - e ^ { 7 } } { 7 }$ 
C)  $\frac { 2 \left( e ^ { 7 \sqrt { 7 } } - e ^ { 7 } \right) } { 3 ( 7 ) ^ { 3 / 2 } }$ 
D)  $\frac { e ^ { 7 \sqrt { 7 } } - e ^ { 7 } } { 14 }$ 
E)  $\frac { 2 \left( e ^ { 7 \sqrt { 7 } } - e ^ { 7 } \right) } { 7 }$ 
A)  $e ^ { 7 \sqrt { 7 } } - e ^ { 7 }$ 
B)  $\frac { e ^ { 7 \sqrt { 7 } } - e ^ { 7 } } { 7 }$ 
C)  $\frac { 2 \left( e ^ { 7 \sqrt { 7 } } - e ^ { 7 } \right) } { 3 ( 7 ) ^ { 3 / 2 } }$ 
D)  $\frac { e ^ { 7 \sqrt { 7 } } - e ^ { 7 } } { 14 }$ 
E)  $\frac { 2 \left( e ^ { 7 \sqrt { 7 } } - e ^ { 7 } \right) } { 7 }$

Question 12

Use sigma notation to write the sum $$\frac { 5 } { 1 + 1 } + \frac { 5 } { 1 + 2 } + \frac { 5 } { 1 + 3 } + \cdots + \frac { 5 } { 1 + 22 }$$

Accepted Answer

A)  $\sum _ { j = 1 } ^ { 22 } \frac { 110 } { 22 + j }$ 
B)  $\sum _ { j = 1 } ^ { 5 } \frac { 110 } { j + 22 }$ 
C)  $\sum _ { j = 1 } ^ { 22 } \frac { 5 } { 22 + j }$ 
D)  $\sum _ { j = 1 } ^ { 22 } \frac { 5 } { 1 + j }$ 
E)  $\sum _ { j = 1 } ^ { 22 } \frac { 110 } { 1 + j }$ 
A)  $\sum _ { j = 1 } ^ { 22 } \frac { 110 } { 22 + j }$ 
B)  $\sum _ { j = 1 } ^ { 5 } \frac { 110 } { j + 22 }$ 
C)  $\sum _ { j = 1 } ^ { 22 } \frac { 5 } { 22 + j }$ 
D)  $\sum _ { j = 1 } ^ { 22 } \frac { 5 } { 1 + j }$ 
E)  $\sum _ { j = 1 } ^ { 22 } \frac { 110 } { 1 + j }$

Question 13

$\text { Find } F ^ { \prime } ( x ) \text { given } \bar { \digamma } ( x ) = \int _ { 1 } ^ { 6 x } \frac { 7 } { t } d t \text {. }$

Accepted Answer

A)  $F F ^ {t } ( x ) = \frac { 13 } { x }$ 
B)  $F ^ { t } ( x ) = \ln | 7 x |$ 
C)  $F ^ { t } ( x ) = \frac { 6 } { x }$ 
D)  $F ^ { t } ( x ) = \ln | 6 x |$ 
E)  $F ^ { t } ( x ) = \frac { 7 } { x }$ 
A)  $F F ^ {t } ( x ) = \frac { 13 } { x }$ 
B)  $F ^ { t } ( x ) = \ln | 7 x |$ 
C)  $F ^ { t } ( x ) = \frac { 6 } { x }$ 
D)  $F ^ { t } ( x ) = \ln | 6 x |$ 
E)  $F ^ { t } ( x ) = \frac { 7 } { x }$

Question 14

$\text { The volume } V \text { in liters of air in the lungs during a five-second respiratory cycle is }$ approximated by the model $V = 0.1724 t + 0.1523 t ^ { 2 } - 0.0377 t ^ { 3 }$ where $t$ is the time in seconds. Approximate the average volume of air in the lungs during one cycle. Round your answer to four decimal places.

Accepted Answer

A)  $\quad 1.8776$ liter 
B)  $2.0094$ liter 
C)  $0.4004$ liter 
D)  $0.8148$ liter 
E)  $0.5220$ liter 
A)  $\quad 1.8776$ liter 
B)  $2.0094$ liter 
C)  $0.4004$ liter 
D)  $0.8148$ liter 
E)  $0.5220$ liter

Question 15

Find the indefinite integral of the following function and check the result by differentiation. $$\int ( 5 - s ) \sqrt { s } d s$$

Accepted Answer

A)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } + \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$ 
B)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 2 } { 5 } s ^ { \frac { 5 } { 2 } } + C$ 
C)  $\frac { 5 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 2 } { 5 } s ^ { \frac { 5 } { 2 } } + C$ 
D)  $\frac { 5 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$ 
E)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$ 
A)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } + \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$ 
B)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 2 } { 5 } s ^ { \frac { 5 } { 2 } } + C$ 
C)  $\frac { 5 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 2 } { 5 } s ^ { \frac { 5 } { 2 } } + C$ 
D)  $\frac { 5 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$ 
E)  $\frac { 10 } { 3 } s ^ { \frac { 3 } { 2 } } - \frac { 5 } { 2 } s ^ { \frac { 5 } { 2 } } + C$

Question 16

Find an equation for the function $f ( x ) \text { whose derivative is } f ^ {t} ( x ) = 4 \sin ( 12 x ) \text { and }$ whose graph passes through the point $\left( \frac { \pi } { 12 } , - \frac { 2 } { 3 } \right)$

Accepted Answer

A)  $f ( x ) = - \frac { 1 } { 24 } \cos ( 3 x ) + 2$ 
B)  $f ( x ) = - \frac { 1 } { 3 } \cos ( 12 x ) - 12$ 
C)  $f ( x ) = \frac { 1 } { 6 } \cos ( 12 x ) + 1$ 
D)  $f ( x ) = \frac { 1 } { 12 } \cos ( 24 x ) + 6$ 
E)  $f ( x ) = - \frac { 1 } { 3 } \cos ( 12 x ) - 1$ 
A)  $f ( x ) = - \frac { 1 } { 24 } \cos ( 3 x ) + 2$ 
B)  $f ( x ) = - \frac { 1 } { 3 } \cos ( 12 x ) - 12$ 
C)  $f ( x ) = \frac { 1 } { 6 } \cos ( 12 x ) + 1$ 
D)  $f ( x ) = \frac { 1 } { 12 } \cos ( 24 x ) + 6$ 
E)  $f ( x ) = - \frac { 1 } { 3 } \cos ( 12 x ) - 1$

Question 17

Find the indefinite integral. $$\int 6 ^ { 5 x } d x$$

Accepted Answer

A)  $\frac { 1 } { \ln 6 } 6 ^ { 5 x } + C$ 
B)  $\frac { 1 } { 5 } 6 ^ { 5 x } + C$ 
C)  $\frac { \ln 6 } { 5 } 6 ^ { 5 x } + C$ 
D)  $\frac { 1 } { 5 \ln 6 } 6 ^ { 5 x } + C$ 
E)  $5 ( \ln 6 ) 6 ^ { 5 x } + C$ 
A)  $\frac { 1 } { \ln 6 } 6 ^ { 5 x } + C$ 
B)  $\frac { 1 } { 5 } 6 ^ { 5 x } + C$ 
C)  $\frac { \ln 6 } { 5 } 6 ^ { 5 x } + C$ 
D)  $\frac { 1 } { 5 \ln 6 } 6 ^ { 5 x } + C$ 
E)  $5 ( \ln 6 ) 6 ^ { 5 x } + C$

Question 18

$\text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. }$

Accepted Answer

A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 5 \pi } { 12 }$ 
E)  $\frac { \pi } { 6 }$ 
A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 5 \pi } { 12 }$ 
E)  $\frac { \pi } { 6 }$

Question 19

Use the Trapezoidal Rule to estimate the number of square meters of land in a lot where $x$ and $y$ are measured in meters as shown in the figure below. The land is bounded by a stream and two straight roads that meet at right angles.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0 & 100 & 200 & 300 & 400 & 500 & 600 & 700 & 800 & 900 & 1000 \
\hline y & 110 & 110 & 105 & 97 & 75 & 75 & 80 & 65 & 60 & 20 & 0 \
\hline
\end{array}$

Accepted Answer

A)  74,200 
B)  74,000 
C)  73,700 
D)  74,700 
E)  $74.400$ 
A)  74,200 
B)  74,000 
C)  73,700 
D)  74,700 
E)  $74.400$

Question 20

Find the indefinite integral $\int e ^ { 4 x ^ { 8 } } x ^ { 7 } d x$

Accepted Answer

A)  $\frac { 1 } { 8 } x ^ { 8 } \ln \left( 4 x ^ { 8 } \right) + C$ 
B)  $\frac { 1 } { 32 } e ^ { 4 x ^ { 9 } } + C$ 
C)  $\frac { 1 } { 32 } e ^ { 4 x ^ { 8 } } + C$ 
D)  $\frac { 1 } { 8 } x ^ { 8 } e ^ { 4 x ^ { 8 } } + C$ 
E)  $\frac { 1 } { 32 } x ^ { 7 } e ^ { 4 x ^ { 8 } } + C$ 
A)  $\frac { 1 } { 8 } x ^ { 8 } \ln \left( 4 x ^ { 8 } \right) + C$ 
B)  $\frac { 1 } { 32 } e ^ { 4 x ^ { 9 } } + C$ 
C)  $\frac { 1 } { 32 } e ^ { 4 x ^ { 8 } } + C$ 
D)  $\frac { 1 } { 8 } x ^ { 8 } e ^ { 4 x ^ { 8 } } + C$ 
E)  $\frac { 1 } { 32 } x ^ { 7 } e ^ { 4 x ^ { 8 } } + C$

Find the following indefinite integral. $\int x ^ { 7 } \left( 4 ^ { - x ^ { 8 } } \right) d x$

Evaluate the definite integral. $\int _ { 1 } ^ { 2 } \frac { 1 - \cos ( 9 \theta ) } { 9 \theta - \sin ( 9 \theta ) } d \theta$

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

Solve the differential equation. $\frac { d f } { d z } = \frac { 4 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) ^ { 2 } }$

Find the sum given below. $\sum _ { k = 3 } ^ { 5 } \frac { 5 } { k ^ { 2 } }$

The graph of the function $g ( y ) = y ^ { 3 }$ is given below. Which of the following definite integrals yields the area of the shaded region? $The graph of the function g ( y ) = y ^ { 3 } is given below. Which of the following definite integrals yields the area of the shaded region?$

Evaluate the definite integral $\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$ . Use a graphing utility to verify your results.

Find the indefinite integral $\int \sqrt [ 13 ] { x ^ { 8 } } d x$

Solve the differential equation. $\frac { d y } { d t } = - 12 t ^ { 2 } , \quad y ( 0 ) = 5$

Evaluate the definite integral. $\int _ { 1 } ^ { 7 } \frac { e ^ { 7 \sqrt { x } } } { \sqrt { x } } d x$

Use sigma notation to write the sum $\frac { 5 } { 1 + 1 } + \frac { 5 } { 1 + 2 } + \frac { 5 } { 1 + 3 } + \cdots + \frac { 5 } { 1 + 22 }$

$\text { Find } F ^ { \prime } ( x ) \text { given } \bar { \digamma } ( x ) = \int _ { 1 } ^ { 6 x } \frac { 7 } { t } d t \text {. }$

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 5 - s ) \sqrt { s } d s$

Find an equation for the function $f ( x ) \text { whose derivative is } f ^ {t} ( x ) = 4 \sin ( 12 x ) \text { and }$ whose graph passes through the point $\left( \frac { \pi } { 12 } , - \frac { 2 } { 3 } \right)$

Find the indefinite integral. $\int 6 ^ { 5 x } d x$

$\text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. }$ $\text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. }$

Find the indefinite integral $\int e ^ { 4 x ^ { 8 } } x ^ { 7 } d x$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

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 5: Antiderivatives and Indefinite Integration

Find the following indefinite integral. ∫x7(4−x8)dx\int x ^ { 7 } \left( 4 ^ { - x ^ { 8 } } \right) d x∫x7(4−x8)dx

Evaluate the definite integral. ∫121−cos⁡(9θ)9θ−sin⁡(9θ)dθ\int _ { 1 } ^ { 2 } \frac { 1 - \cos ( 9 \theta ) } { 9 \theta - \sin ( 9 \theta ) } d \theta∫12​9θ−sin(9θ)1−cos(9θ)​dθ

Find the indefinite integral ∫3x2cos⁡x3dx\int 3 x ^ { 2 } \cos x ^ { 3 } d x∫3x2cosx3dx

Solve the differential equation. dfdz=4z+7(2z2+7z+2)2\frac { d f } { d z } = \frac { 4 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) ^ { 2 } }dzdf​=(2z2+7z+2)24z+7​

Find the sum given below. ∑k=355k2\sum _ { k = 3 } ^ { 5 } \frac { 5 } { k ^ { 2 } }∑k=35​k25​

The graph of the function g(y)=y3g ( y ) = y ^ { 3 }g(y)=y3 is given below. Which of the following definite integrals yields the area of the shaded region?

Evaluate the definite integral ∫374u3du\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u∫37​u34​du . Use a graphing utility to verify your results.

Find the indefinite integral ∫x813dx\int \sqrt [ 13 ] { x ^ { 8 } } d x∫13x8​dx

Solve the differential equation. dydt=−12t2,y(0)=5\frac { d y } { d t } = - 12 t ^ { 2 } , \quad y ( 0 ) = 5dtdy​=−12t2,y(0)=5

Evaluate the definite integral. ∫17e7xxdx\int _ { 1 } ^ { 7 } \frac { e ^ { 7 \sqrt { x } } } { \sqrt { x } } d x∫17​x​e7x​​dx

Use sigma notation to write the sum 51+1+51+2+51+3+⋯+51+22\frac { 5 } { 1 + 1 } + \frac { 5 } { 1 + 2 } + \frac { 5 } { 1 + 3 } + \cdots + \frac { 5 } { 1 + 22 }1+15​+1+25​+1+35​+⋯+1+225​

Find F′(x) given ϝˉ(x)=∫16x7tdt. \text { Find } F ^ { \prime } ( x ) \text { given } \bar { \digamma } ( x ) = \int _ { 1 } ^ { 6 x } \frac { 7 } { t } d t \text {. } Find F′(x) given ϝˉ​(x)=∫16x​t7​dt.

Find the indefinite integral of the following function and check the result by differentiation. ∫(5−s)sds\int ( 5 - s ) \sqrt { s } d s∫(5−s)s​ds

Find the indefinite integral. ∫65xdx\int 6 ^ { 5 x } d x∫65xdx

Find the area of the shaded region for the function y=ex1+e2x. \text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. } Find the area of the shaded region for the function y=1+e2xex​.

Find the indefinite integral ∫e4x8x7dx\int e ^ { 4 x ^ { 8 } } x ^ { 7 } d x∫e4x8x7dx

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

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 following indefinite integral. $\int x ^ { 7 } \left( 4 ^ { - x ^ { 8 } } \right) d x$

Evaluate the definite integral. $\int _ { 1 } ^ { 2 } \frac { 1 - \cos ( 9 \theta ) } { 9 \theta - \sin ( 9 \theta ) } d \theta$

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

Solve the differential equation. $\frac { d f } { d z } = \frac { 4 z + 7 } { \left( 2 z ^ { 2 } + 7 z + 2 \right) ^ { 2 } }$

Find the sum given below. $\sum _ { k = 3 } ^ { 5 } \frac { 5 } { k ^ { 2 } }$

The graph of the function $g ( y ) = y ^ { 3 }$ is given below. Which of the following definite integrals yields the area of the shaded region? $The graph of the function g ( y ) = y ^ { 3 } is given below. Which of the following definite integrals yields the area of the shaded region?$

Evaluate the definite integral $\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$ . Use a graphing utility to verify your results.

Find the indefinite integral $\int \sqrt [ 13 ] { x ^ { 8 } } d x$

Solve the differential equation. $\frac { d y } { d t } = - 12 t ^ { 2 } , \quad y ( 0 ) = 5$

Evaluate the definite integral. $\int _ { 1 } ^ { 7 } \frac { e ^ { 7 \sqrt { x } } } { \sqrt { x } } d x$

Use sigma notation to write the sum $\frac { 5 } { 1 + 1 } + \frac { 5 } { 1 + 2 } + \frac { 5 } { 1 + 3 } + \cdots + \frac { 5 } { 1 + 22 }$

$\text { Find } F ^ { \prime } ( x ) \text { given } \bar { \digamma } ( x ) = \int _ { 1 } ^ { 6 x } \frac { 7 } { t } d t \text {. }$

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 5 - s ) \sqrt { s } d s$

Find the indefinite integral. $\int 6 ^ { 5 x } d x$

$\text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. }$ $\text { Find the area of the shaded region for the function } y = \frac { e ^ { x } } { 1 + e ^ { 2 x } } \text {. }$

Find the indefinite integral $\int e ^ { 4 x ^ { 8 } } x ^ { 7 } d x$