Question 1

Use the Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$. Round the result to four decimal places.

Accepted Answer

The answer of Use the Trapezoidal Rule to approximate \(\int...

Question 2

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 3

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 4

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

The answer of Evaluate the integral using the indicated trigonometric...

Question 5

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 6

Determine whether the improper integral converges or diverges, and if it converges, find its value.
$$\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$$

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Find the integral.
$$\int \frac { d x } { x ( x - 2 ) }$$

Accepted Answer

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

Question 9

Use the Table of Integrals to evaluate the integral.
$$\int e ^ { 6 x } \sin 3 x d x$$

Accepted Answer

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

Question 10

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 11

Use a table of integrals to evaluate the integral.
$$\int e ^ { - 7 x } \sin 3 x d x$$

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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 15

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

Accepted Answer

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

Question 16

Evaluate the integral.
$$\int _ { 2 } ^ { \infty } \frac { d y } { y ^ { 2 } + 2 y - 3 }$$

Accepted Answer

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

Question 17

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

A)  $20.28$ 
B)  $17.89$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $15.87$ 
A)  $20.28$ 
B)  $17.89$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $15.87$

Question 18

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 19

Find the integral using an appropriate trigonometric substitution.
$$\int x \sqrt { 16 - x ^ { 2 } } d x$$

Accepted Answer

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

Question 20

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

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

Use the Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$ . Round the result to four decimal places.

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

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.

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

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

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$

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

Find the integral. $\int \frac { d x } { x ( x - 2 ) }$

Use the Table of Integrals to evaluate the integral. $\int e ^ { 6 x } \sin 3 x d x$

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

Use a table of integrals to evaluate the integral. $\int e ^ { - 7 x } \sin 3 x d x$

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

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

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

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

Evaluate the integral. $\int _ { 2 } ^ { \infty } \frac { d y } { y ^ { 2 } + 2 y - 3 }$

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

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 16 - x ^ { 2 } } d x$

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

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

Use the Trapezoidal Rule to approximate ∫23e3/xdx\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x∫23​e3/xdx for n=4n = 4n=4 . Round the result to four decimal places.

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

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.

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

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

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫π/25π/6cos⁡x1−sin⁡xdx\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x∫π/25π/6​1−sinx​cosx​dx

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

Find the integral. ∫dxx(x−2)\int \frac { d x } { x ( x - 2 ) }∫x(x−2)dx​

Use the Table of Integrals to evaluate the integral. ∫e6xsin⁡3xdx\int e ^ { 6 x } \sin 3 x d x∫e6xsin3xdx

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

Use a table of integrals to evaluate the integral. ∫e−7xsin⁡3xdx\int e ^ { - 7 x } \sin 3 x d x∫e−7xsin3xdx

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

Evaluate the integral. ∫x/3x/22cot⁡2xdx\int _ { x / 3 } ^ { x / 2 } 2 \cot ^ { 2 } x d x∫x/3x/2​2cot2xdx

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

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

Evaluate the integral. ∫2∞dyy2+2y−3\int _ { 2 } ^ { \infty } \frac { d y } { y ^ { 2 } + 2 y - 3 }∫2∞​y2+2y−3dy​

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.

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​

Find the integral using an appropriate trigonometric substitution. ∫x16−x2dx\int x \sqrt { 16 - x ^ { 2 } } d x∫x16−x2​dx

Find the integral. ∫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

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 Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$ . Round the result to four decimal places.

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

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.

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

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

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$

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

Find the integral. $\int \frac { d x } { x ( x - 2 ) }$

Use the Table of Integrals to evaluate the integral. $\int e ^ { 6 x } \sin 3 x d x$

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

Use a table of integrals to evaluate the integral. $\int e ^ { - 7 x } \sin 3 x d x$

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

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

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

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

Evaluate the integral. $\int _ { 2 } ^ { \infty } \frac { d y } { y ^ { 2 } + 2 y - 3 }$

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

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 16 - x ^ { 2 } } d x$

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