Question 1

Find the indefinite integral.
$$\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$$

Accepted Answer

A)  $\frac { 3 } { 7 } x ^ { 7 } - 2 x ^ { 4 } + 5 x + C$ 
B)  $5 x - 8 x ^ { 4 } + 3 x ^ { 7 } + C$ 
C)  $\frac { 3 } { 7 } x ^ { 6 } - 2 x ^ { 3 } + 5 x + C$ 
D)  $5 x - 24 x ^ { 2 } + 18 x ^ { 5 } + C$ 
A)  $\frac { 3 } { 7 } x ^ { 7 } - 2 x ^ { 4 } + 5 x + C$ 
B)  $5 x - 8 x ^ { 4 } + 3 x ^ { 7 } + C$ 
C)  $\frac { 3 } { 7 } x ^ { 6 } - 2 x ^ { 3 } + 5 x + C$ 
D)  $5 x - 24 x ^ { 2 } + 18 x ^ { 5 } + C$

Question 2

$$\text { Evaluate the integral } \int _ { 0 } ^ { a / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$$

Accepted Answer

The answer of \[\text { Evaluate the integral } \int...

Question 3

Evaluate the definite integral.
$$\int _ { 0 } ^ { \pi / 8 } \sin 7 t d t$$

Accepted Answer

The answer of Evaluate the definite integral.
\[\int _ { 0...

Question 4

Evaluate the indefinite integral.
$$\int \sec ^ { 2 } x \tan x d x$$

Accepted Answer

The answer of Evaluate the indefinite integral.
\[\int \sec ^ {...

Question 5

Find the integral using an appropriate trigonometric substitution.
$$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$

Question 6

$$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$$

Accepted Answer

A sketch of the area between t

Question 7

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

Accepted Answer

The answer of Find the integral.
\[\int x ^ { 4...

Question 8

Evaluate
$$\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$$

Accepted Answer

The answer of Evaluate
\[\frac { d } { d x...

Question 9

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the
table. Use the Midpoint Rule to estimate the distance traveled by the car. $$\begin{array} { | c | c | c | c | c | } 
\hline t ( \mathrm {~s} ) & v ( \mathrm { mi } / \mathrm { h } ) & & t ( \mathrm {~s} ) & v ( \mathrm { mi } / \mathrm { h } ) \
\hline 0 & 0 & & 60 & 54 \
\hline 10 & 34 & & 70 & 67 \
\hline 20 & 43 & & 80 & 77 \
\hline 30 & 36 & & 90 & 37 \
\hline 40 & 45 & & 100 & 42 \
\hline 50 & 45 & & & \
\hline
\end{array}$$

Accepted Answer

The answer of The velocity of a car was read...

Question 10

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

Accepted Answer

A)  191 
B)  188 
C)  190 
D)  189 
E)  192 
A)  191 
B)  188 
C)  190 
D)  189 
E)  192

Question 11

Use the Midpoint Rule with $n = 5$ to approximate the integral.
$$\int _ { 0 } ^ { 10 } 5 \sin \sqrt { q } d q$$
Round your answer to three decimal places.

Accepted Answer

The answer of Use the Midpoint Rule with \(n =...

Question 12

Find the indefinite integral.
$$\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$$

Accepted Answer

A)  $2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$ 
A)  $2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$

Question 13

Use the Midpoint Rule with $n = 5$ to approximate the integral.
$$\int _ { 0 } ^ { 10 } 2 \sin \sqrt { q } d q$$
Round your answer to three decimal places. Select the correct answer.

Accepted Answer

A)  $18.864$ 
B)  $18.064$ 
C)  $17.064$ 
D)  $17.564$ 
E)  $12.564$ 
A)  $18.864$ 
B)  $18.064$ 
C)  $17.064$ 
D)  $17.564$ 
E)  $12.564$

Question 14

Evaluate the indefinite integral.
$$\int 7 e ^ { \cos x } \sin x d x$$

Accepted Answer

A)  $- e ^ { \cos x } \sin x + C$ 
B)  $- 7 e ^ { \cos x } + C$ 
C)  $e ^ { 7 \sin x } + C$ 
D)  $7 e ^ { \cos x } \sin x + C$ 
E)  $- 7 \sin \left( e ^ { \cos x } \right) + C$ 
A)  $- e ^ { \cos x } \sin x + C$ 
B)  $- 7 e ^ { \cos x } + C$ 
C)  $e ^ { 7 \sin x } + C$ 
D)  $7 e ^ { \cos x } \sin x + C$ 
E)  $- 7 \sin \left( e ^ { \cos x } \right) + C$

Question 15

Evaluate the limit after first finding the sum (as a function of $n$ ) using the summation formulas. $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 4 + \frac { 2 k } { n } \right) ^ { 2 } \left( \frac { 1 } { n } \right)$

Accepted Answer

The answer of Evaluate the limit after first finding the...

Question 16

Find the integral.
$$\int e ^ { 3 x } \sin 5 x d x$$
Select the correct answer.

Accepted Answer

A)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) - C$ 
B)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 3 x ) - C$ 
C)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) + C$ 
D)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 5 x ) + C$ 
A)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) - C$ 
B)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 3 x ) - C$ 
C)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) + C$ 
D)  $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 5 x ) + C$

Question 17

Find the indefinite integral.
$$\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$$

Accepted Answer

The answer of Find the indefinite integral.
\[\int \frac { 2...

Question 18

Find the integral using an appropriate trigonometric substitution.
$$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$$

Accepted Answer

The answer of Find the integral using an appropriate trigonometric...

Question 19

Find the integral.
$$\int \sin ^ { 3 } x \cos ^ { 6 } x d x$$

Accepted Answer

A)  $\frac { 1 } { 7 } \cos ^ { 7 } x - \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
B)  $- \frac { 1 } { 9 } \cos ^ { 9 } x + \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
C)  $- \frac { 1 } { 7 } \cos ^ { 7 } x + \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
D)  $\frac { 1 } { 9 } \cos ^ { 9 } x - \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
A)  $\frac { 1 } { 7 } \cos ^ { 7 } x - \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
B)  $- \frac { 1 } { 9 } \cos ^ { 9 } x + \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
C)  $- \frac { 1 } { 7 } \cos ^ { 7 } x + \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
D)  $\frac { 1 } { 9 } \cos ^ { 9 } x - \frac { 1 } { 7 } \cos ^ { 7 } x + C$

Question 20

Find the integral using the indicated substitution.
$$\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$$

Accepted Answer

The answer of Find the integral using the indicated substitution.
\[\int...

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

$\text { Evaluate the integral } \int _ { 0 } ^ { a / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 7 t d t$

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$

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

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t() v(/) t() v(/) 0 0 60 54 10 34 70 67 20 43 80 77 30 36 90 37 40 45 100 42 50 45

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

Use the Midpoint Rule with $n = 5$ to approximate the integral. $\int _ { 0 } ^ { 10 } 5 \sin \sqrt { q } d q$ Round your answer to three decimal places.

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Use the Midpoint Rule with $n = 5$ to approximate the integral. $\int _ { 0 } ^ { 10 } 2 \sin \sqrt { q } d q$ Round your answer to three decimal places. Select the correct answer.

Evaluate the indefinite integral. $\int 7 e ^ { \cos x } \sin x d x$

Evaluate the limit after first finding the sum (as a function of $n$ ) using the summation formulas. $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 4 + \frac { 2 k } { n } \right) ^ { 2 } \left( \frac { 1 } { n } \right)$

Find the integral. $\int e ^ { 3 x } \sin 5 x d x$ Select the correct answer.

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

Find the integral. $\int \sin ^ { 3 } x \cos ^ { 6 } x d x$

Find the integral using the indicated substitution. $\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

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Exam 4: Applications of Differentiation

Find the indefinite integral. ∫(5−8x3+3x6)dx\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x∫(5−8x3+3x6)dx

Evaluate the integral ∫0a/20etan⁡5xcos⁡25xdx\text { Evaluate the integral } \int _ { 0 } ^ { a / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x Evaluate the integral ∫0a/20​cos25xetan5x​dx

Evaluate the definite integral. ∫0π/8sin⁡7tdt\int _ { 0 } ^ { \pi / 8 } \sin 7 t d t∫0π/8​sin7tdt

Evaluate the indefinite integral. ∫sec⁡2xtan⁡xdx\int \sec ^ { 2 } x \tan x d x∫sec2xtanxdx

Find the integral using an appropriate trigonometric substitution. ∫x3x2+36dx\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x∫x2+36​x3​dx

Show that ∫0x−5cos⁡2xdx=0 by interpreting the definite integral geometrically. \text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. } Show that ∫0x​−5cos2xdx=0 by interpreting the definite integral geometrically.

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

Evaluate ddx∫0x(earcsin⁡t)dt\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d tdxd​∫0x​(earcsint)dt

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t() v(/) t() v(/) 0 0 60 54 10 34 70 67 20 43 80 77 30 36 90 37 40 45 100 42 50 45

Estimate the area from 0 to 5 under the graph of f(x)=49−x2f ( x ) = 49 - x ^ { 2 }f(x)=49−x2 using five approximating rectangles and right endpoints.

Use the Midpoint Rule with n=5n = 5n=5 to approximate the integral. ∫0105sin⁡qdq\int _ { 0 } ^ { 10 } 5 \sin \sqrt { q } d q∫010​5sinq​dq Round your answer to three decimal places.

Find the indefinite integral. ∫2x7−5x4+2x7dx\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x∫x72x7−5x4+2​dx

Use the Midpoint Rule with n=5n = 5n=5 to approximate the integral. ∫0102sin⁡qdq\int _ { 0 } ^ { 10 } 2 \sin \sqrt { q } d q∫010​2sinq​dq Round your answer to three decimal places. Select the correct answer.

Evaluate the indefinite integral. ∫7ecos⁡xsin⁡xdx\int 7 e ^ { \cos x } \sin x d x∫7ecosxsinxdx

Find the integral. ∫e3xsin⁡5xdx\int e ^ { 3 x } \sin 5 x d x∫e3xsin5xdx Select the correct answer.

Find the indefinite integral. ∫2x7−5x4+2x7dx\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x∫x72x7−5x4+2​dx

Find the integral using an appropriate trigonometric substitution. ∫x3x2+36dx\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x∫x2+36​x3​dx

Find the integral. ∫sin⁡3xcos⁡6xdx\int \sin ^ { 3 } x \cos ^ { 6 } x d x∫sin3xcos6xdx

Find the integral using the indicated substitution. ∫tan⁡5xsec⁡2xdx,u=tan⁡x\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x∫tan5xsec2xdx,u=tanx

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

$\text { Evaluate the integral } \int _ { 0 } ^ { a / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 7 t d t$

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$

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

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

Use the Midpoint Rule with $n = 5$ to approximate the integral. $\int _ { 0 } ^ { 10 } 5 \sin \sqrt { q } d q$ Round your answer to three decimal places.

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Use the Midpoint Rule with $n = 5$ to approximate the integral. $\int _ { 0 } ^ { 10 } 2 \sin \sqrt { q } d q$ Round your answer to three decimal places. Select the correct answer.

Evaluate the indefinite integral. $\int 7 e ^ { \cos x } \sin x d x$

Find the integral. $\int e ^ { 3 x } \sin 5 x d x$ Select the correct answer.

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

Find the integral. $\int \sin ^ { 3 } x \cos ^ { 6 } x d x$

Find the integral using the indicated substitution. $\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$