Question 1

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 2

Evaluate the integral
$$\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$$

Accepted Answer

The answer of Evaluate the integral
\[\int \frac { 6 }...

Question 3

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

Accepted Answer

The answer of Use long division to evaluate the integral.
\[\int...

Question 4

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 5

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 6

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

Accepted Answer

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

Question 7

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 8

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

Accepted Answer

The answer of Find the integral.
\[\int \cot ^ { 2...

Question 9

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

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 10

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

Accepted Answer

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

Question 11

Evaluate the integral.
$\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

Accepted Answer

A)  $\frac { 5 } { 32 } \pi$ 
B)  $\frac { 3 } { 16 } \pi$ 
C)  $\frac { 5 } { 16 } \pi$ 
D)  $\frac { 3 } { 32 } \pi$ 
A)  $\frac { 5 } { 32 } \pi$ 
B)  $\frac { 3 } { 16 } \pi$ 
C)  $\frac { 5 } { 16 } \pi$ 
D)  $\frac { 3 } { 32 } \pi$

Question 12

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

Accepted Answer

The answer of \[\text { Find the area bounded by...

Question 13

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

Accepted Answer

A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$

Question 14

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 15

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

Accepted Answer

A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$

Question 16

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 17

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

Accepted Answer

The answer of Determine whether the improper integral converges or...

Question 18

Evaluate the integral. Select the correct answer.
$\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Accepted Answer

A)  $2.8665$ 
B)  $6.5700$ 
C)  $0.7166$ 
D)  $11.8665$ 
E)  None of these 
A)  $2.8665$ 
B)  $6.5700$ 
C)  $0.7166$ 
D)  $11.8665$ 
E)  None of these

Question 19

Evaluate the integral.
$$\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$$

Accepted Answer

A)  $4 \left( \frac { 1 } { 3 } x - \frac { 1 } { 3 } \ln x \right) + C$ 
B)  $4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$ 
C)  $- 4 \left( \frac { \sqrt { 2 } } { 2 } \arctan x \right) + C$ 
D)  $- 4 ( \ln x - x ) + C$ 
E)  None of these 
A)  $4 \left( \frac { 1 } { 3 } x - \frac { 1 } { 3 } \ln x \right) + C$ 
B)  $4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$ 
C)  $- 4 \left( \frac { \sqrt { 2 } } { 2 } \arctan x \right) + C$ 
D)  $- 4 ( \ln x - x ) + C$ 
E)  None of these

Question 20

The region $\left\{ ( x + y ) \mid x \geq - 7,0 \leq y \leq e ^ { - x / 5 } \right\}$ is represented below. Find the area of this region to two decimal places.

Accepted Answer

The answer of The region \(\left\{ ( x + y...

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 $\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$

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

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

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 using an appropriate trigonometric substitution. $\int x \sqrt { 1 - x ^ { 2 } } d x$

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

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

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

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

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

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

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

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

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

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

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

Evaluate the integral. Select the correct answer. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral. $\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$

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

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Exam 7: Techniques of Integration

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 ∫6e3x−exdx\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x∫e3x−ex6​dx

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

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 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 using an appropriate trigonometric substitution. ∫x1−x2dx\int x \sqrt { 1 - x ^ { 2 } } d x∫x1−x2​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

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

Evaluate the integral using integration by parts with the indicated choices of uuu and dvd vdv . ∫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

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

Evaluate the integral. ∫0π/2cos⁡6xdx\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x∫0π/2​cos6xdx

Use a table of integrals to evaluate the integral. ∫x2+2xdx\int x \sqrt { 2 + 2 x } d x∫x2+2x​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

Use a table of integrals to evaluate the integral. ∫x2+2xdx\int x \sqrt { 2 + 2 x } d x∫x2+2x​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

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫3∞1x3dx\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 3 } } d x∫3∞​x31​dx

Evaluate the integral. Select the correct answer. ∫019xsin⁡πxdx\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x∫01​9xsinπxdx

Evaluate the integral. ∫4(x−1x2+2x)dx\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x∫4(x2+2xx−1​)dx

The region {(x+y)∣x≥−7,0≤y≤e−x/5}\left\{ ( x + y ) \mid x \geq - 7,0 \leq y \leq e ^ { - x / 5 } \right\}{(x+y)∣x≥−7,0≤y≤e−x/5} is represented below. Find the area of this region to two decimal places.

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

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 $\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$

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

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

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 using an appropriate trigonometric substitution. $\int x \sqrt { 1 - x ^ { 2 } } d x$

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

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

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

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

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

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

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

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

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

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

Evaluate the integral. Select the correct answer. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral. $\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$