Question 1

Evaluate the integral. $$\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x$$

Accepted Answer

$$8 \ln \frac { 4 } { \sqrt { 17 } + 1 } + \frac { \sqrt { 17 } } { 2 }$$

Question 2

Evaluate the integral using an appropriate trigonometric substitution. $$\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$$

Accepted Answer

$$\frac { 16 \sqrt { 2 } } { 81 }$$

Question 3

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $$\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$$

Accepted Answer

A)  $2.0076$ 
B)  $2.9504$ 
C)  $2.6098$ 
D)  $2.2955$ 
A)  $2.0076$ 
B)  $2.9504$ 
C)  $2.6098$ 
D)  $2.2955$ D

Question 4

Estimate the area of the shaded region by using the Trapezoidal Rule with $n = 2$. Round the answer to the nearest tenth.

Accepted Answer

The answer of Estimate the area of the shaded region...

Question 5

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 6

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 7

Use the Table of Integrals to evaluate the integral. $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 6 } - 4 } } d x$$

Accepted Answer

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

Question 8

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 9

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

Accepted Answer

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

Question 10

Evaluate the integral using integration by parts with the indicated choices of u and dv. $$\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$$

Accepted Answer

A)  $6 \sin \theta + 6 \cos \theta + C$ 
B)  $6 \theta \cos \theta + 6 \sin \theta + C$ 
C)  $6 \theta \sin \theta + 6 \cos \theta + C$ 
D)  $6 \sin \theta - 6 \cos \theta + C$ 
E)  None of these 
A)  $6 \sin \theta + 6 \cos \theta + C$ 
B)  $6 \theta \cos \theta + 6 \sin \theta + C$ 
C)  $6 \theta \sin \theta + 6 \cos \theta + C$ 
D)  $6 \sin \theta - 6 \cos \theta + C$ 
E)  None of these

Question 11

Use the Table of Integrals to evaluate the integral to three decimal places. $$\int _ { 3 } ^ { 5 } \frac { 1 } { x ^ { 2 } \sqrt { 4 x ^ { 2 } - 1 } } d x$$

Accepted Answer

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

Question 12

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Accepted Answer

A)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 5 \sqrt { x ^ { 2 } + 25 } + C$ 
B)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 25 \sqrt { x ^ { 2 } + 25 } + C$ 
E)  $\frac { 3 } { 2 } ( x + 25 ) ^ { 3 / 2 } - 25 \sqrt { x + 25 } + C$ 
A)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 5 \sqrt { x ^ { 2 } + 25 } + C$ 
B)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 25 \sqrt { x ^ { 2 } + 25 } + C$ 
E)  $\frac { 3 } { 2 } ( x + 25 ) ^ { 3 / 2 } - 25 \sqrt { x + 25 } + C$

Question 13

Evaluate the integral. $$\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$$

Accepted Answer

A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these 
A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these

Question 14

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Accepted Answer

The answer of Find a bound on the error in...

Question 15

Find the integral. $$\int \cot ^ { 2 } x \csc ^ { 6 } x d x$$

Accepted Answer

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

Question 16

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 17

Evaluate the integral. $$\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x$$

Accepted Answer

A)  $12 - 8 \ln 12 + \ln \sqrt { 8 }$ 
B)  $\frac { 1 } { 8 \ln 12 } - \ln \sqrt { 8 }$ 
C)  $4 - 12 \ln 8 + \sqrt { 8 } \ln 16$ 
D)  $4 - 8 ( \ln 12 - \ln 8 )$ 
E)  None of these 
A)  $12 - 8 \ln 12 + \ln \sqrt { 8 }$ 
B)  $\frac { 1 } { 8 \ln 12 } - \ln \sqrt { 8 }$ 
C)  $4 - 12 \ln 8 + \sqrt { 8 } \ln 16$ 
D)  $4 - 8 ( \ln 12 - \ln 8 )$ 
E)  None of these

Question 18

Find the integral. $$\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$$

Accepted Answer

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

Question 19

Determine whether the improper integral converges or diverges, and if it converges, find its value. $$\int _ { - 27 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } d x$$

Accepted Answer

A)  15 
B)  $- \frac { 15 } { 2 }$ 
C)  Diverges 
D)  $\frac { 15 } { 2 }$ 
A)  15 
B)  $- \frac { 15 } { 2 }$ 
C)  Diverges 
D)  $\frac { 15 } { 2 }$

Question 20

Evaluate the integral. $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } + 6 x ^ { 2 } + 7 x + 2 } { ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x$$

Accepted Answer

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

Evaluate the integral. $\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$

Estimate the area of the shaded region by using the Trapezoidal Rule with $n = 2$ . Round the answer to the nearest tenth.

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

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

Use the Table of Integrals to evaluate the integral. $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 6 } - 4 } } d x$

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.

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

Evaluate the integral using integration by parts with the indicated choices of u and dv. $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Use the Table of Integrals to evaluate the integral to three decimal places. $\int _ { 3 } ^ { 5 } \frac { 1 } { x ^ { 2 } \sqrt { 4 x ^ { 2 } - 1 } } d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Evaluate the integral. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Find the integral. $\int \cot ^ { 2 } x \csc ^ { 6 } x d x$

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

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x$

Find the integral. $\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { - 27 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } + 6 x ^ { 2 } + 7 x + 2 } { ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x$

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

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Further Applications of Integration

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Exam 7: Techniques of Integration

Evaluate the integral. ∫417x2−16dx\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x∫417​​x2−16​dx

Evaluate the integral using an appropriate trigonometric substitution. ∫13x2−1x4dx\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x∫13​x4x2−1​​dx

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. ∫−11x2+1dx;n=6\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6∫−11​x2+1​dx;n=6

Estimate the area of the shaded region by using the Trapezoidal Rule with n=2n = 2n=2 . Round the answer to the nearest tenth.

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

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

Use the Table of Integrals to evaluate the integral. ∫x2x6−4dx\int \frac { x ^ { 2 } } { \sqrt { x ^ { 6 } - 4 } } d x∫x6−4​x2​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.

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

Evaluate the integral using integration by parts with the indicated choices of u and dv. ∫6θcos⁡θdθ,u=6θ,dv=cos⁡θdθ\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta∫6θcosθdθ,u=6θ,dv=cosθdθ

Use the Table of Integrals to evaluate the integral to three decimal places. ∫351x24x2−1dx\int _ { 3 } ^ { 5 } \frac { 1 } { x ^ { 2 } \sqrt { 4 x ^ { 2 } - 1 } } d x∫35​x24x2−1​1​dx

Evaluate the integral using the indicated trigonometric substitution. ∫x3x2+25dx;x=5tan⁡θ\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta∫x2+25​x3​dx;x=5tanθ

Evaluate the integral. ∫08(x2+8)e−xdx\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x∫08​(x2+8)e−xdx

Find a bound on the error in approximating the integral ∫19ln⁡x9\int _ { 1 } ^ { 9 } \ln x ^ { 9 }∫19​lnx9 using (a) the Trapezoidal Rule and (b) Simpson's Rule with n=10n = 10n=10 subintervals.

Find the integral. ∫cot⁡2xcsc⁡6xdx\int \cot ^ { 2 } x \csc ^ { 6 } x d x∫cot2xcsc6xdx

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 integral. ∫04xx+8dx\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x∫04​x+8x​dx

Find the integral. ∫3x2−7x−2(x2+x+1)2dx\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x∫(x2+x+1)23x2−7x−2​dx

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫−2781x3dx\int _ { - 27 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } d x∫−278​3x​1​dx

Evaluate the integral. ∫013x3+6x2+7x+2(x+1)2(x2+1)dx\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } + 6 x ^ { 2 } + 7 x + 2 } { ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x∫01​(x+1)2(x2+1)3x3+6x2+7x+2​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

Evaluate the integral. $\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$

Estimate the area of the shaded region by using the Trapezoidal Rule with $n = 2$ . Round the answer to the nearest tenth.

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

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

Use the Table of Integrals to evaluate the integral. $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 6 } - 4 } } d x$

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.

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

Evaluate the integral using integration by parts with the indicated choices of u and dv. $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Use the Table of Integrals to evaluate the integral to three decimal places. $\int _ { 3 } ^ { 5 } \frac { 1 } { x ^ { 2 } \sqrt { 4 x ^ { 2 } - 1 } } d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Evaluate the integral. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Find the integral. $\int \cot ^ { 2 } x \csc ^ { 6 } x d x$

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

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x$

Find the integral. $\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { - 27 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } + 6 x ^ { 2 } + 7 x + 2 } { ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x$