Question 1

Find the integral.
$\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$

Accepted Answer

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

Question 2

The region under the graph of
$$y = \frac { 7 } { x ( x + 1 ) }$$
on the interval $[ 1,2 ]$ is revolved about the $x$-axis. Find the volume of the resulting solid.

Accepted Answer

The answer of The region under the graph of
\[y =...

Question 3

Evaluate the integral or show that it is divergent.
$$\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$$

Accepted Answer

A)  $- \frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 5 } { 4 } \pi$ 
E)  divergent 
A)  $- \frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 5 } { 4 } \pi$ 
E)  divergent

Question 4

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value.
$$\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$$

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM&
Ex

Question 5

Evaluate the integral.
$$
\int_{1}^{\infty} \frac{d x}{x \ln x}
$$

Accepted Answer

A)  $ \frac{1}{2} $ 
B)  $ -\frac{1}{2} $ 
C)  $ -\frac{1}{4} $ 
D)  $ \frac{1}{4} $ 
E)  divergent 
A)  $ \frac{1}{2} $ 
B)  $ -\frac{1}{2} $ 
C)  $ -\frac{1}{4} $ 
D)  $ \frac{1}{4} $ 
E)  divergent

Question 6

Find the integral.
$$\int \frac { 3 x - 9 } { x ^ { 2 } - 4 x - 5 } d x$$

Accepted Answer

The answer of Find the integral.
\[\int \frac { 3 x...

Question 7

Evaluate the integral or show that it is divergent.
$$\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$$

Accepted Answer

The answer of Evaluate the integral or show that it...

Question 8

Use long division to evaluate the integral.
$$\int \frac { x ^ { 2 } } { x + 7 } d x$$

Accepted Answer

A)  $\frac { 1 } { 2 } ( x - 21 ) ( x + 7 ) + 49 \ln | x + 7 | + C$ 
B)  $\frac { 1 } { 2 } ( x + 21 ) ( x + 7 ) - 49 \ln | x + 7 | + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - 14 x - 147 + \ln | x + 7 | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - 14 x + 147 + 49 \ln | x + 7 | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } + 2 x + \ln | x + 49 | + C$ 
A)  $\frac { 1 } { 2 } ( x - 21 ) ( x + 7 ) + 49 \ln | x + 7 | + C$ 
B)  $\frac { 1 } { 2 } ( x + 21 ) ( x + 7 ) - 49 \ln | x + 7 | + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - 14 x - 147 + \ln | x + 7 | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - 14 x + 147 + 49 \ln | x + 7 | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } + 2 x + \ln | x + 49 | + C$

Question 9

Make a substitution to express the integrand as a rational function and then evaluate the integral.
$$\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$$
Round the answer to four decimal places.

Accepted Answer

The answer of Make a substitution to express the integrand...

Question 10

Use long division to evaluate the integral.
$$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$$
The choices are rounded to 3 decimal places.

Accepted Answer

A)  $- 3.544$ 
B)  $6.456$ 
C)  $1.456$ 
D)  $- 8.544$ 
E)  $11.456$ 
A)  $- 3.544$ 
B)  $6.456$ 
C)  $1.456$ 
D)  $- 8.544$ 
E)  $11.456$

Question 11

Use a table of integrals to evaluate the integral.
$$\int x ^ { 3 } \sin \left( x ^ { 2 } + 10 \right) d x$$

Accepted Answer

The answer of Use a table of integrals to evaluate...

Question 12

Find the integral. Select the correct answer.
$$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$$

Accepted Answer

A)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 2 } { x - 1 } + x + C$ 
B)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 2 } { x - 1 } + x + C$ 
C)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 1 } { x - 1 } + x + C$ 
D)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 1 } { x - 1 } + x + C$ 
A)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 2 } { x - 1 } + x + C$ 
B)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 2 } { x - 1 } + x + C$ 
C)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 1 } { x - 1 } + x + C$ 
D)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 1 } { x - 1 } + x + C$

Question 13

Evaluate the integral.
$$\int \frac { 7 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int \frac { 7 \sin...

Question 14

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$.
$$f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$$

Accepted Answer

The answer of Find the average value of the function...

Question 15

Use the Table of Integrals to evaluate the integral.
$$\int \frac { \sqrt { 25 x ^ { 2 } - 1 } } { x ^ { 2 } } d x$$

Accepted Answer

The answer of Use the Table of Integrals to evaluate...

Question 16

Find the area bounded by the curves $y = \cos x$ and $y = \cos ^ { 2 } x$ between $x = 0$ and $x = \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  none of these 
A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  none of these

Question 17

Use a table of integrals to evaluate the integral.
$\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x$

Accepted Answer

A) 
$- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
B) 
$- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
C) 
$- \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
D) 
$- \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
A) 
$- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
B) 
$- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
C) 
$- \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$ 
D) 
$- \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$

Question 18

Evaluate the indefinite integral.
$$\int x \cos 9 x d x$$

Accepted Answer

The answer of Evaluate the indefinite integral.
\[\int x \cos 9...

Question 19

Evaluate the integral to six decimal places.
$$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$$

Accepted Answer

The answer of Evaluate the integral to six decimal places.
\[\int...

Question 20

Find the integral using an appropriate trigonometric substitution.
$$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } + 36 } } d x$$

Accepted Answer

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

Find the integral. $\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$

The region under the graph of $y = \frac { 7 } { x ( x + 1 ) }$ on the interval $[ 1,2 ]$ is revolved about the $x$ -axis. Find the volume of the resulting solid.

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. $\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x \ln x}$

Find the integral. $\int \frac { 3 x - 9 } { x ^ { 2 } - 4 x - 5 } d x$

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Use long division to evaluate the integral. $\int \frac { x ^ { 2 } } { x + 7 } d x$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Use a table of integrals to evaluate the integral. $\int x ^ { 3 } \sin \left( x ^ { 2 } + 10 \right) d x$

Find the integral. Select the correct answer. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Evaluate the integral. $\int \frac { 7 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Use the Table of Integrals to evaluate the integral. $\int \frac { \sqrt { 25 x ^ { 2 } - 1 } } { x ^ { 2 } } d x$

Find the area bounded by the curves $y = \cos x$ and $y = \cos ^ { 2 } x$ between $x = 0$ and $x = \frac { \pi } { 2 }$ .

Use a table of integrals to evaluate the integral. $\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x$

Evaluate the indefinite integral. $\int x \cos 9 x d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } + 36 } } d x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications 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 7: Techniques of Integration

Find the integral. ∫3x−5x2−2x−3dx\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x∫x2−2x−33x−5​dx

The region under the graph of y=7x(x+1)y = \frac { 7 } { x ( x + 1 ) }y=x(x+1)7​ on the interval [1,2][ 1,2 ][1,2] is revolved about the xxx -axis. Find the volume of the resulting solid.

Evaluate the integral or show that it is divergent. ∫−∞∞5dx4x2+4x+5\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }∫−∞∞​4x2+4x+55dx​

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. ∫0x3sin⁡xdx;n=4\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4∫0x​3sinxdx;n=4

Evaluate the integral. ∫1∞dxxln⁡x\int_{1}^{\infty} \frac{d x}{x \ln x}∫1∞​xlnxdx​

Find the integral. ∫3x−9x2−4x−5dx\int \frac { 3 x - 9 } { x ^ { 2 } - 4 x - 5 } d x∫x2−4x−53x−9​dx

Evaluate the integral or show that it is divergent. ∫−∞∞5dx4x2+4x+5\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }∫−∞∞​4x2+4x+55dx​

Use long division to evaluate the integral. ∫x2x+7dx\int \frac { x ^ { 2 } } { x + 7 } d x∫x+7x2​dx

Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫425xx−100dx\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x∫425​x−100x​​dx Round the answer to four decimal places.

Use long division to evaluate the integral. ∫01x3+12x2−36x+1x2+4x−12dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x∫01​x2+4x−12x3+12x2−36x+1​dx The choices are rounded to 3 decimal places.

Use a table of integrals to evaluate the integral. ∫x3sin⁡(x2+10)dx\int x ^ { 3 } \sin \left( x ^ { 2 } + 10 \right) d x∫x3sin(x2+10)dx

Find the integral. Select the correct answer. ∫x3−3x2+6x−2x3−2x2+xdx\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x∫x3−2x2+xx3−3x2+6x−2​dx

Evaluate the integral. ∫7sin⁡2x1+cos⁡4xdx\int \frac { 7 \sin 2 x } { 1 + \cos ^ { 4 } x } d x∫1+cos4x7sin2x​dx

Find the average value of the function f(x)f ( x )f(x) in the interval [−π,π][ - \pi , \pi ][−π,π] . f(x)=sin⁡6xcos⁡5xf ( x ) = \sin ^ { 6 } x \cos ^ { 5 } xf(x)=sin6xcos5x

Use the Table of Integrals to evaluate the integral. ∫25x2−1x2dx\int \frac { \sqrt { 25 x ^ { 2 } - 1 } } { x ^ { 2 } } d x∫x225x2−1​​dx

Find the area bounded by the curves y=cos⁡xy = \cos xy=cosx and y=cos⁡2xy = \cos ^ { 2 } xy=cos2x between x=0x = 0x=0 and x=π2x = \frac { \pi } { 2 }x=2π​ .

Use a table of integrals to evaluate the integral. ∫6x+5x2dx\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x∫x26x+5​​dx

Evaluate the indefinite integral. ∫xcos⁡9xdx\int x \cos 9 x d x∫xcos9xdx

Evaluate the integral to six decimal places. ∫01x364−x2dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x∫01​64−x2​x3​dx

Find the integral using an appropriate trigonometric substitution. ∫1x2x2+36dx\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } + 36 } } d x∫x2x2+36​1​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications 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. $\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$

The region under the graph of $y = \frac { 7 } { x ( x + 1 ) }$ on the interval $[ 1,2 ]$ is revolved about the $x$ -axis. Find the volume of the resulting solid.

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. $\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$

Evaluate the integral. $\int_{1}^{\infty} \frac{d x}{x \ln x}$

Find the integral. $\int \frac { 3 x - 9 } { x ^ { 2 } - 4 x - 5 } d x$

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Use long division to evaluate the integral. $\int \frac { x ^ { 2 } } { x + 7 } d x$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Use a table of integrals to evaluate the integral. $\int x ^ { 3 } \sin \left( x ^ { 2 } + 10 \right) d x$

Find the integral. Select the correct answer. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Evaluate the integral. $\int \frac { 7 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Use the Table of Integrals to evaluate the integral. $\int \frac { \sqrt { 25 x ^ { 2 } - 1 } } { x ^ { 2 } } d x$

Find the area bounded by the curves $y = \cos x$ and $y = \cos ^ { 2 } x$ between $x = 0$ and $x = \frac { \pi } { 2 }$ .

Use a table of integrals to evaluate the integral. $\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x$

Evaluate the indefinite integral. $\int x \cos 9 x d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } + 36 } } d x$