Question 1

Find the derivative of the function. $$g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$$

Accepted Answer

The answer of Find the derivative of the function. \[g...

Question 2

$$\text { Find a function } f ( x ) \text { such that } 5 + \int _ { a } ^ { x } \frac { f ( t ) } { t ^ { 2 } } d t = 12 \sqrt { x } \text { for } x > 0 \text { and some number } a \text {. }$$

Accepted Answer

The answer of \[\text { Find a function } f...

Question 3

Evaluate the integral by making the given substitution. $$\int \cos 3 x d x , u = 3 x$$

Accepted Answer

The answer of Evaluate the integral by making the given...

Question 4

Find the indefinite integral. $$\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$$

Accepted Answer

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

Question 5

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline w & 0 & 1 & 2 & 3 & 4 & 5 & 6 \
\hline f(w) & 9.7 & 9.1 & 7.7 & 6.1 & 4.2 & -6.6 & -10.3 \
\hline
\end{array}$

Accepted Answer

The answer of The table gives the values of a...

Question 6

$$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$$

Accepted Answer

The answer of \[\text { Find the interval on which...

Question 7

Find the area of the region that lies beneath the given curve. $$y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$$

Accepted Answer

A)  $1.500$ 
B)  $1.450$ 
C)  $- 0.500$ 
D)  $- 1.500$ 
E)  $0.500$ 
A)  $1.500$ 
B)  $1.450$ 
C)  $- 0.500$ 
D)  $- 1.500$ 
E)  $0.500$

Question 8

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 9

Find the integral using an appropriate trigonometric substitution. $$\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$$

Accepted Answer

A)  $- \sqrt { 2 - x } + C$ 
B)  $\sqrt { 2 - x } + C$ 
C)  $\sqrt { 4 - x ^ { 2 } } + C$ 
D)  $- \sqrt { 4 - x ^ { 2 } } + C$ 
A)  $- \sqrt { 2 - x } + C$ 
B)  $\sqrt { 2 - x } + C$ 
C)  $\sqrt { 4 - x ^ { 2 } } + C$ 
D)  $- \sqrt { 4 - x ^ { 2 } } + C$

Question 10

Find the general indefinite integral. $$\int \frac { \sin 18 t } { \sin 9 t } d t$$

Accepted Answer

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

Question 11

If $h ^ { t }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Accepted Answer

A)  $\int _ { 3 } ^ { 7 } h ^ { t } ( t ) d t$ 
B)  $h ^ { t} ( 7 ) - h ^ { t } ( 3 )$ 
C)  $\int _ { 7 } ^ { 3 } h ( t ) d t$ 
A)  $\int _ { 3 } ^ { 7 } h ^ { t } ( t ) d t$ 
B)  $h ^ { t} ( 7 ) - h ^ { t } ( 3 )$ 
C)  $\int _ { 7 } ^ { 3 } h ( t ) d t$

Question 12

Find the indefinite integral. $$\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$$

Accepted Answer

A)  $- 2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
B)  $2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
C)  $\left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
A)  $- 2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
B)  $2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
C)  $\left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$

Question 13

Find the derivative of the function. $$g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$$

Accepted Answer

The answer of Find the derivative of the function. \[g...

Question 14

Evaluate the integral.
$\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Accepted Answer

A)  $- 0.500$ 
B)  $- 1.000$ 
C)  $- 0.400$ 
D)  $0.250$ 
E)  $1.000$ 
A)  $- 0.500$ 
B)  $- 1.000$ 
C)  $- 0.400$ 
D)  $0.250$ 
E)  $1.000$

Question 15

Find the indefinite integral.
$$\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$$

Accepted Answer

A)  $2 x + \frac { 1 } { x ^ { 6 } } - \frac { 3 } { 8 x ^ { 8 } } + C$ 
B) 
$2 x + \frac { 3 } { 4 x ^ { 8 } } - \frac { 3 } { 10 x ^ { 10 } } + C$ 
C) 
$2 x + x ^ { 6 } - \frac { 3 } { 8 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 3 } { 4 } x ^ { 8 } - \frac { 3 } { 10 } x ^ { 10 } + C$ 
A)  $2 x + \frac { 1 } { x ^ { 6 } } - \frac { 3 } { 8 x ^ { 8 } } + C$ 
B) 
$2 x + \frac { 3 } { 4 x ^ { 8 } } - \frac { 3 } { 10 x ^ { 10 } } + C$ 
C) 
$2 x + x ^ { 6 } - \frac { 3 } { 8 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 3 } { 4 } x ^ { 8 } - \frac { 3 } { 10 } x ^ { 10 } + C$

Question 16

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Accepted Answer

The answer of The graph of a function $f$ on...

Question 17

Evaluate the integral if it exists. $$\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$$

Accepted Answer

A)  $1 - 10 \ln x + C$ 
B) $x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D) $x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these 
A)  $1 - 10 \ln x + C$ 
B) $x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D) $x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these

Question 18

Find the area of the region under the graph of $f$ on $[ a , b ]$.
$$f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$$

Accepted Answer

A)  $- 3$ 
B)  $- 6$ 
C)  6
d 3 
A)  $- 3$ 
B)  $- 6$ 
C)  6
d 3

Question 19

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Accepted Answer

The answer of Approximate the area under the curve \(y...

Question 20

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

Accepted Answer

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

Find the derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

$\text { Find a function } f ( x ) \text { such that } 5 + \int _ { a } ^ { x } \frac { f ( t ) } { t ^ { 2 } } d t = 12 \sqrt { x } \text { for } x > 0 \text { and some number } a \text {. }$

Evaluate the integral by making the given substitution. $\int \cos 3 x d x , u = 3 x$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$

Find the area of the region that lies beneath the given curve. $y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$

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

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

If $h ^ { t }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Find the indefinite integral. $\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

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

Functions and Models

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Integrals

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Techniques of Integration

Further Applications of Integration

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Exam 4: Applications of Differentiation

Find the derivative of the function. g(x)=∫cos⁡x9xcos⁡(t3)dtg ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d tg(x)=∫cosx9x​cos(t3)dt

Evaluate the integral by making the given substitution. ∫cos⁡3xdx,u=3x\int \cos 3 x d x , u = 3 x∫cos3xdx,u=3x

Find the indefinite integral. ∫x−2/3x1/3−2dx\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x∫x−2/3x1/3−2​dx

The table gives the values of a function obtained from an experiment. Use the values to estimate ∫06f(w)dw\int _ { 0 } ^ { 6 } f ( w ) d w∫06​f(w)dw using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

Find the interval on which the curve F(x)=∫0xdt11+11t is concave downward. \text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. } Find the interval on which the curve F(x)=∫0x​11+11tdt​ is concave downward.

Find the area of the region that lies beneath the given curve. y=sin⁡x,0≤x≤π3y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }y=sinx,0≤x≤3π​

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

Find the integral using an appropriate trigonometric substitution. ∫x4−x2dx\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x∫4−x2​x​dx

Find the general indefinite integral. ∫sin⁡18tsin⁡9tdt\int \frac { \sin 18 t } { \sin 9 t } d t∫sin9tsin18t​dt

If hth ^ { t }ht is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Find the indefinite integral. ∫(x2+4x−5)2(2x+4)dx\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x∫(x2+4x−5)2(2x+4)dx

Find the derivative of the function. g(x)=∫4x241+t2dtg ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d tg(x)=∫4x2​41+t2​dt

Evaluate the integral. ∫x/3x/7sin⁡tdt\int _ { x / 3 } ^ { x / 7 } \sin t d t∫x/3x/7​sintdt

Find the indefinite integral. ∫2x9−6x2+3x9dx\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x∫x92x9−6x2+3​dx

Evaluate the integral if it exists. ∫(5−xx)2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​)2dx

Find the area of the region under the graph of fff on [a,b][ a , b ][a,b] . f(x)=x2−2x+2;[−1,2]f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]f(x)=x2−2x+2;[−1,2]

Approximate the area under the curve y=5x2y = \frac { 5 } { x ^ { 2 } }y=x25​ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

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

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 derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

Evaluate the integral by making the given substitution. $\int \cos 3 x d x , u = 3 x$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$

Find the area of the region that lies beneath the given curve. $y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$

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

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

If $h ^ { t }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Find the indefinite integral. $\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

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