Question 1

$$\text { Find the indefinite integral } \int \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 16 } d x \text {. }$$

Accepted Answer

The answer of \[\text { Find the indefinite integral }...

Question 2

Evaluate the integral.
$$\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$$

Accepted Answer

A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 10 }$ 
C)  $\frac { \pi } { 24 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $10 \pi$ 
A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 10 }$ 
C)  $\frac { \pi } { 24 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $10 \pi$

Question 3

Use the graph of $f$ shown in the figure to evaluate the integral by interpreting it geometrically.

$$\int _ { 2 } ^ { 7 } f ( x ) d x$$

Accepted Answer

The answer of Use the graph of $f$ shown in...

Question 4

If $h ^ { \prime }$ 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

The answer of If $h ^ { \prime }$ is...

Question 5

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$$

Accepted Answer

The answer of Use Part 1 of the Fundamental Theorem...

Question 6

Evaluate the integral by making the given substitution.
$$\int \cos 2 x d x , u = 2 x$$

Accepted Answer

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

Question 7

Evaluate the integral.
$$\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$$

Accepted Answer

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

Question 8

Evaluate the integral.
$$\int _ { - 2 } ^ { 5 } \left| 4 x - x ^ { 2 } \right| d x$$
The choices are rounded to the nearest hundredth.

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { - 2...

Question 9

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

Accepted Answer

The answer of Find the integral.
\[\int \cos ^ { 3...

Question 10

Evaluate the integral.
$$\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$$

Accepted Answer

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

Question 11

You are given function $f$ defined on an interval $[ a , b ]$, the number $n$ of subintervals of equal length $\Delta x = \frac { ( b - a ) } { n }$, and the evaluation points $c _ { k }$ in $\left[ x _ { k - 1 } , x _ { k } \right]$.
(a) Sketch the graph of $f$ and the rectangles associated with the Riemann sum for $f$ on $[ a , b ]$, and
(b) find the Riemann sum.
$f ( x ) = 2 x - 3 , \quad [ 0,2 ] , \quad n = 4 , \quad c _ { k }$ is the right endpoint

Accepted Answer

The answer of You are given function $f$ defined on...

Question 12

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 13

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 14

Find the integral using an appropriate trigonometric substitution.
$$\int x \sqrt { 9 - x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
B)  $- \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
C)  $- \frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
D)  $\frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
A)  $\frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
B)  $- \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
C)  $- \frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
D)  $\frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$

Question 15

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

Accepted Answer

The answer of Find the integral.
\[\int \sin ^ { 3...

Question 16

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 17

Evaluate the integral if it exists.
$$\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$$

Accepted Answer

The answer of Evaluate the integral if it exists.
\[\int _...

Question 18

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval $[ a , b ]$.
$$\lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { 5 } } \sum _ { k = 1 } ^ { n } k ^ { 4 } ; \quad [ 0,3 ]$$

Accepted Answer

The answer of Evaluate the limit by interpreting it as...

Question 19

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

Accepted Answer

The answer of Find the interval on which the curve...

Question 20

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 (...

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 16 } d x \text {. }$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Use the graph of $f$ shown in the figure to evaluate the integral by interpreting it geometrically. $Use the graph of f shown in the figure to evaluate the integral by interpreting it geometrically. \int _ { 2 } ^ { 7 } f ( x ) d x$ $\int _ { 2 } ^ { 7 } f ( x ) d x$

If $h ^ { \prime }$ 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 ?

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

Evaluate the integral by making the given substitution. $\int \cos 2 x d x , u = 2 x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Evaluate the integral. $\int _ { - 2 } ^ { 5 } \left| 4 x - x ^ { 2 } \right| d x$ The choices are rounded to the nearest hundredth.

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

Evaluate the integral. $\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$

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

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

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 9 - x ^ { 2 } } d x$

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

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

Evaluate the integral if it exists. $\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval $[ a , b ]$ . $\lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { 5 } } \sum _ { k = 1 } ^ { n } k ^ { 4 } ; \quad [ 0,3 ]$

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

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

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Exam 4: Applications of Differentiation

Find the indefinite integral ∫x2−16x2+16dx. \text { Find the indefinite integral } \int \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 16 } d x \text {. } Find the indefinite integral ∫x2+16x2−16​dx.

Evaluate the integral. ∫06136+t2dt\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t∫06​36+t21​dt

Use the graph of fff shown in the figure to evaluate the integral by interpreting it geometrically. ∫27f(x)dx\int _ { 2 } ^ { 7 } f ( x ) d x∫27​f(x)dx

If h′h ^ { \prime }h′ 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 ?

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1xz2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x)=∫1x​​z4+1z2​dz

Evaluate the integral by making the given substitution. ∫cos⁡2xdx,u=2x\int \cos 2 x d x , u = 2 x∫cos2xdx,u=2x

Evaluate the integral. ∫01/24dr1−r2\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }∫01/2​1−r2​4dr​

Evaluate the integral. ∫−25∣4x−x2∣dx\int _ { - 2 } ^ { 5 } \left| 4 x - x ^ { 2 } \right| d x∫−25​​4x−x2​dx The choices are rounded to the nearest hundredth.

Find the integral. ∫cos⁡3xsin⁡2xdx\int \cos ^ { 3 } x \sin ^ { 2 } x d x∫cos3xsin2xdx

Evaluate the integral. ∫017x6cos⁡(x7)dx\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x∫01​7x6cos(x7)dx

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

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 integral using an appropriate trigonometric substitution. ∫x9−x2dx\int x \sqrt { 9 - x ^ { 2 } } d x∫x9−x2​dx

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

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

Evaluate the integral if it exists. ∫013x2cos⁡(x2)dx\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x∫01​3x2cos(x2)dx

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a,b][ a , b ][a,b] . lim⁡n→∞1n5∑k=1nk4;[0,3]\lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { 5 } } \sum _ { k = 1 } ^ { n } k ^ { 4 } ; \quad [ 0,3 ]limn→∞​n51​∑k=1n​k4;[0,3]

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

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

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

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 16 } d x \text {. }$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Use the graph of $f$ shown in the figure to evaluate the integral by interpreting it geometrically. $Use the graph of f shown in the figure to evaluate the integral by interpreting it geometrically. \int _ { 2 } ^ { 7 } f ( x ) d x$ $\int _ { 2 } ^ { 7 } f ( x ) d x$

If $h ^ { \prime }$ 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 ?

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

Evaluate the integral by making the given substitution. $\int \cos 2 x d x , u = 2 x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Evaluate the integral. $\int _ { - 2 } ^ { 5 } \left| 4 x - x ^ { 2 } \right| d x$ The choices are rounded to the nearest hundredth.

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

Evaluate the integral. $\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$

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

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

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 9 - x ^ { 2 } } d x$

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

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

Evaluate the integral if it exists. $\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval $[ a , b ]$ . $\lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { 5 } } \sum _ { k = 1 } ^ { n } k ^ { 4 } ; \quad [ 0,3 ]$

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

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