Question 1

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

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 2

Evaluate the integral.
$$\int _ { x / 3 } ^ { x / 2 } \sin t d t$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { x /...

Question 3

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 4

Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { x / 4 } 6 \sin x d x$ :
If $m \leq f ( x ) \leq M$ on $[ a , b ]$, then
$$m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$$

Accepted Answer

The answer of Use the following property of the definite...

Question 5

Evaluate the definite integral.
$$\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$$

Accepted Answer

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

Question 6

The area of the region that lies to the right of the $y$-axis and to the left of the parabola $x = 2 y - y ^ { 2 }$ (the shaded region in the figure) is given by the integral $\int _ { 0 } ^ { 2 } \left( 2 y - y ^ { 2 } \right) d y$.
Find the area..

Accepted Answer

The answer of The area of the region that lies...

Question 7

Evaluate the integral.
$$\int _ { x / 2 } ^ { \pi } \cos \left( \frac { 1 } { 2 } x \right) d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { x /...

Question 8

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 9

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

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$

Question 10

Evaluate the integral by making the given substitution.
$$\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$$

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$

Question 12

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

Accepted Answer

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

Question 13

Evaluate the integral.
$\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } } d x$
Select the correct answer.

Accepted Answer

A)  $- 5$ 
B)  8 
C)  4 
D)  $- 8$ 
A)  $- 5$ 
B)  8 
C)  4 
D)  $- 8$

Question 14

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

Accepted Answer

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

Question 15

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 16

Evaluate the definite integral.
$$\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$$

Accepted Answer

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

Question 17

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$, find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Accepted Answer

A)  7 
B)  $- 13$ 
C)  20 
D)  6 
E)  13 
A)  7 
B)  $- 13$ 
C)  20 
D)  6 
E)  13

Question 18

Find the area of the region under the graph of $f$ on $[ a , b ]$.
$$f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$$

Accepted Answer

A)  3 
B)  $- 12$ 
C)  12 
D)  $- 3$ 
A)  3 
B)  $- 12$ 
C)  12 
D)  $- 3$

Question 19

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { s / 4 } 4 \sin x d x$ :
If $m \leq f ( x ) \leq M$ on $[ a , b ]$, then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Accepted Answer

The answer of Use the following property of the definite...

Question 20

Find the integral using the indicated substitution.
$$\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$$

Accepted Answer

A)  $\frac { 1 } { 28 } ( 4 x + 3 ) ^ { 7 } + C$ 
B)  $6 ( 4 x + 3 ) ^ { 7 } + C$ 
C)  $\frac { 1 } { 6 } ( 4 x + 3 ) ^ { 7 } + C$ 
D)  $\frac { 1 } { 7 } ( 4 x + 3 ) ^ { 7 } + C$ 
A)  $\frac { 1 } { 28 } ( 4 x + 3 ) ^ { 7 } + C$ 
B)  $6 ( 4 x + 3 ) ^ { 7 } + C$ 
C)  $\frac { 1 } { 6 } ( 4 x + 3 ) ^ { 7 } + C$ 
D)  $\frac { 1 } { 7 } ( 4 x + 3 ) ^ { 7 } + C$

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

Evaluate the integral. $\int _ { x / 3 } ^ { x / 2 } \sin t d t$

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

Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { x / 4 } 6 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$

Evaluate the integral. $\int _ { x / 2 } ^ { \pi } \cos \left( \frac { 1 } { 2 } x \right) d x$

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

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

Evaluate the integral by making the given substitution. $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$

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

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

Evaluate the integral. $\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } } d x$ Select the correct answer.

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

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

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { s / 4 } 4 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Find the integral using the indicated substitution. $\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$

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

Exam 4: Applications of Differentiation

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

Evaluate the integral. ∫x/3x/2sin⁡tdt\int _ { x / 3 } ^ { x / 2 } \sin t d t∫x/3x/2​sintdt

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

Evaluate the definite integral. ∫0π/8sin⁡5tdt\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t∫0π/8​sin5tdt

Evaluate the integral. ∫x/2πcos⁡(12x)dx\int _ { x / 2 } ^ { \pi } \cos \left( \frac { 1 } { 2 } x \right) d x∫x/2π​cos(21​x)dx

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​

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 by making the given substitution. ∫x2x3+2dx,u=x3+2\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2∫x2x3+2​dx,u=x3+2

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

Find the derivative of the function. g(x)=∫1x1t2+2dtg ( x ) = \int _ { 1 } ^ { x } \frac { 1 } { t ^ { 2 } + 2 } d tg(x)=∫1x​t2+21​dt

Evaluate the integral. ∫191xdx\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } } d x∫19​x​1​dx Select the correct answer.

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. 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

Evaluate the definite integral. ∫0π/8sin⁡5tdt\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t∫0π/8​sin5tdt

If ∫06f(x)dx=13\int _ { 0 } ^ { 6 } f ( x ) d x = 13∫06​f(x)dx=13 and ∫04f(x)dx=7\int _ { 0 } ^ { 4 } f ( x ) d x = 7∫04​f(x)dx=7 , find ∫46f(x)dx\int _ { 4 } ^ { 6 } f ( x ) d x∫46​f(x)dx

Find the area of the region under the graph of fff on [a,b][ a , b ][a,b] . f(x)=x2−2x+4;[−1,2]f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]f(x)=x2−2x+4;[−1,2]

Find the integral using the indicated substitution. ∫(4x+3)6dx,u=4x+3\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3∫(4x+3)6dx,u=4x+3

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

Evaluate the integral. $\int _ { x / 3 } ^ { x / 2 } \sin t d t$

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

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$

Evaluate the integral. $\int _ { x / 2 } ^ { \pi } \cos \left( \frac { 1 } { 2 } x \right) d x$

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

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

Evaluate the integral by making the given substitution. $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$

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

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

Evaluate the integral. $\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } } d x$ Select the correct answer.

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

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

Evaluate the definite integral. $\int _ { 0 } ^ { \pi / 8 } \sin 5 t d t$

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$

Find the integral using the indicated substitution. $\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$