Question 1

Evaluate the integral. $\int x \sin 3 x \cos 3 x d x$

Accepted Answer

A)  $-\frac{1}{9}(\cos 9 x-9 x \sin 9 x)+C$ 
B)  $\frac{1}{18}(\sin 3 x-3 x \cos 3 x)+C$ 
C)  $\frac{1}{9}(\sin 9 x-49 \cos 9 x)+C$ 
D)  $-\frac{1}{18}(\sin 18 x-18 x \cos 18 x)+C$ 
E)  None of these 
A)  $-\frac{1}{9}(\cos 9 x-9 x \sin 9 x)+C$ 
B)  $\frac{1}{18}(\sin 3 x-3 x \cos 3 x)+C$ 
C)  $\frac{1}{9}(\sin 9 x-49 \cos 9 x)+C$ 
D)  $-\frac{1}{18}(\sin 18 x-18 x \cos 18 x)+C$ 
E)  None of these B

Question 2

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

$0.018$

Question 3

Find the integral using an appropriate trigonometric substitution. $\int \frac{\sqrt{25 x^{2}-16}}{x} d x$

Accepted Answer

$\sqrt{25 x^{2}-16}-4 \sec ^{-1}\left(\frac{5}{4} x\right)+C$

Question 4

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

Accepted Answer

A)  $\frac{5}{2}$ 
B)  $\frac{5}{4}$ 
C)  $\frac{1}{\ln 4}$ 
D)  $\text { divergent }$ 
E)  $\frac{\ln 5}{4}$ 
A)  $\frac{5}{2}$ 
B)  $\frac{5}{4}$ 
C)  $\frac{1}{\ln 4}$ 
D)  $\text { divergent }$ 
E)  $\frac{\ln 5}{4}$

Question 5

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 x^{2}-7 x-2}{\left(x^{2}+x+1\right)^{2}}...

Question 6

Evaluate the integral. $\int_{0}^{x / 12} 5 \cos ^{5} 6 x d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{0}^{x / 12} 5...

Question 7

Evaluate the integral. $\int \frac{1}{-e^{-5 x}+e^{5 x}} d x$

Accepted Answer

A)  $-\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
B)  $\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
C)  $\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
D)  $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}}\right)+C$ 
E)  $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
A)  $-\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
B)  $\frac{1}{10} \ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
C)  $\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$ 
D)  $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}}\right)+C$ 
E)  $-\ln \left(\frac{\left|e^{5 x}-1\right|}{e^{5 x}+1}\right)+C$

Question 8

Find the integral using an appropriate trigonometric substitution. $\int \frac{1}{x^{2} \sqrt{x^{2}+25}} d x$

Accepted Answer

A)  $-\frac{\sqrt{x^{2}+25}}{5 x}+C$ 
B)  $\frac{\sqrt{x^{2}+25}}{5 x}+C$ 
C)  $\frac{\sqrt{x^{2}+25}}{25 x}+C$ 
D)  $-\frac{\sqrt{x^{2}+25}}{25 x}+C$ 
A)  $-\frac{\sqrt{x^{2}+25}}{5 x}+C$ 
B)  $\frac{\sqrt{x^{2}+25}}{5 x}+C$ 
C)  $\frac{\sqrt{x^{2}+25}}{25 x}+C$ 
D)  $-\frac{\sqrt{x^{2}+25}}{25 x}+C$

Question 9

Evaluate the integral. $\int 15 x \ln (1+x) d x$

Accepted Answer

A)  $15\left(\frac{1}{6} x^{2}-\frac{1}{9} x^{3}+\frac{x^{3}}{3} \ln (1+x)+\frac{1}{3} \ln (1+x)\right)+C$ 
B)  $\frac{15}{4}\left(2\left(x^{2}-1\right) \ln (1+x)-x(x-2)\right)+C$ 
C)  $15\left(\frac{1}{4} x^{2}-\frac{1}{2} \ln (1+x)+\frac{x^{2}}{2}(\ln (1+x))\right)+C$ 
D)  $\frac{15}{4}\left(2\left(x^{2}-1\right) \ln (1+x)+x(x-2)\right)+C$ 
E)  $15\left(-\frac{1}{3} x+\frac{1}{6} x^{2}-\frac{1}{9} x^{4}+\frac{1}{3} \ln (1+x)\right)+C$ 
A)  $15\left(\frac{1}{6} x^{2}-\frac{1}{9} x^{3}+\frac{x^{3}}{3} \ln (1+x)+\frac{1}{3} \ln (1+x)\right)+C$ 
B)  $\frac{15}{4}\left(2\left(x^{2}-1\right) \ln (1+x)-x(x-2)\right)+C$ 
C)  $15\left(\frac{1}{4} x^{2}-\frac{1}{2} \ln (1+x)+\frac{x^{2}}{2}(\ln (1+x))\right)+C$ 
D)  $\frac{15}{4}\left(2\left(x^{2}-1\right) \ln (1+x)+x(x-2)\right)+C$ 
E)  $15\left(-\frac{1}{3} x+\frac{1}{6} x^{2}-\frac{1}{9} x^{4}+\frac{1}{3} \ln (1+x)\right)+C$

Question 10

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

A)  Diverges 
B)  $\frac{1}{9}$ 
C)  $\frac{1}{18}$ 
D)  0 
A)  Diverges 
B)  $\frac{1}{9}$ 
C)  $\frac{1}{18}$ 
D)  0

Question 11

Use a table of integrals to evaluate the integral. $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$

Accepted Answer

A)  $-\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
B)  $-\frac{\sqrt{4 x+5}}{x}+\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
C)  $-\frac{\sqrt{4 x+5}}{x}+\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
D)  $-\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
A)  $-\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
B)  $-\frac{\sqrt{4 x+5}}{x}+\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
C)  $-\frac{\sqrt{4 x+5}}{x}+\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$ 
D)  $-\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$

Question 12

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 13

Evaluate the integral. $\int_{-1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{e^{\arctan y}}{1+y^{2}} d y$

Accepted Answer

The answer of Evaluate the integral. \(\int_{-1 / \sqrt{3}}^{1 /...

Question 14

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

Accepted Answer

A)  $2.8665$ 
B)  6.5700 
C)  0.7166 
D)  11.8665 
E)  $\text { None of these }$ 
A)  $2.8665$ 
B)  6.5700 
C)  0.7166 
D)  11.8665 
E)  $\text { None of these }$

Question 15

Evaluate the integral. $\int \frac{\cos (\ln 18 t)}{t} d t$

Accepted Answer

The answer of Evaluate the integral. \(\int \frac{\cos (\ln 18...

Question 16

Evaluate the integral if it is convergent. $\int_{0}^{\infty} e^{-3 x} d x$

Accepted Answer

The answer of Evaluate the integral if it is convergent....

Question 17

Use long division to evaluate the integral. $\int \frac{x^{2}}{x+3} d x$

Accepted Answer

A)  $\frac{1}{2}(x-9)(x+3)+9 \ln |x+3|+C$ 
B)  $\frac{1}{2}(x+9)(x+3)-9 \ln |x+3|+C$ 
C)  $\frac{x^{2}}{2}-6 x-27+\ln |x+3|+C$ 
D)  $\frac{x^{2}}{2}-6 x+27+9 \ln |x+3|+C$ 
E)  $\frac{x^{2}}{2}+2 x+\ln |x+9|+C$ 
A)  $\frac{1}{2}(x-9)(x+3)+9 \ln |x+3|+C$ 
B)  $\frac{1}{2}(x+9)(x+3)-9 \ln |x+3|+C$ 
C)  $\frac{x^{2}}{2}-6 x-27+\ln |x+3|+C$ 
D)  $\frac{x^{2}}{2}-6 x+27+9 \ln |x+3|+C$ 
E)  $\frac{x^{2}}{2}+2 x+\ln |x+9|+C$

Question 18

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 19

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

Question 20

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)  $\frac{3}{32} \pi$

Evaluate the integral. $\int x \sin 3 x \cos 3 x d x$

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

Find the integral using an appropriate trigonometric substitution. $\int \frac{\sqrt{25 x^{2}-16}}{x} d x$

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

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

Evaluate the integral. $\int_{0}^{x / 12} 5 \cos ^{5} 6 x d x$

Evaluate the integral. $\int \frac{1}{-e^{-5 x}+e^{5 x}} d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac{1}{x^{2} \sqrt{x^{2}+25}} d x$

Evaluate the integral. $\int 15 x \ln (1+x) d x$

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

Use a table of integrals to evaluate the integral. $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$

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. $\int_{-1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{e^{\arctan y}}{1+y^{2}} d y$

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

Evaluate the integral. $\int \frac{\cos (\ln 18 t)}{t} d t$

Evaluate the integral if it is convergent. $\int_{0}^{\infty} e^{-3 x} d x$

Use long division to evaluate the integral. $\int \frac{x^{2}}{x+3} d x$

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

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

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

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Exam 7: Techniques of Integration

Evaluate the integral. ∫xsin⁡3xcos⁡3xdx\int x \sin 3 x \cos 3 x d x∫xsin3xcos3xdx

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

Find the integral using an appropriate trigonometric substitution. ∫25x2−16xdx\int \frac{\sqrt{25 x^{2}-16}}{x} d x∫x25x2−16​​dx

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

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

Evaluate the integral. ∫0x/125cos⁡56xdx\int_{0}^{x / 12} 5 \cos ^{5} 6 x d x∫0x/12​5cos56xdx

Evaluate the integral. ∫1−e−5x+e5xdx\int \frac{1}{-e^{-5 x}+e^{5 x}} d x∫−e−5x+e5x1​dx

Find the integral using an appropriate trigonometric substitution. ∫1x2x2+25dx\int \frac{1}{x^{2} \sqrt{x^{2}+25}} d x∫x2x2+25​1​dx

Evaluate the integral. ∫15xln⁡(1+x)dx\int 15 x \ln (1+x) d x∫15xln(1+x)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

Use a table of integrals to evaluate the integral. ∫4x+5x2dx\int \frac{\sqrt{4 x+5}}{x^{2}} d x∫x24x+5​​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. ∫−1/31/3earctan⁡y1+y2dy\int_{-1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{e^{\arctan y}}{1+y^{2}} d y∫−1/3​1/3​​1+y2earctany​dy

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

Evaluate the integral. ∫cos⁡(ln⁡18t)tdt\int \frac{\cos (\ln 18 t)}{t} d t∫tcos(ln18t)​dt

Evaluate the integral if it is convergent. ∫0∞e−3xdx\int_{0}^{\infty} e^{-3 x} d x∫0∞​e−3xdx

Use long division to evaluate the integral. ∫x2x+3dx\int \frac{x^{2}}{x+3} d x∫x+3x2​dx

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 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​

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

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Evaluate the integral. $\int x \sin 3 x \cos 3 x d x$

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

Find the integral using an appropriate trigonometric substitution. $\int \frac{\sqrt{25 x^{2}-16}}{x} d x$

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

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

Evaluate the integral. $\int_{0}^{x / 12} 5 \cos ^{5} 6 x d x$

Evaluate the integral. $\int \frac{1}{-e^{-5 x}+e^{5 x}} d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac{1}{x^{2} \sqrt{x^{2}+25}} d x$

Evaluate the integral. $\int 15 x \ln (1+x) d x$

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

Use a table of integrals to evaluate the integral. $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$

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. $\int_{-1 / \sqrt{3}}^{1 / \sqrt{3}} \frac{e^{\arctan y}}{1+y^{2}} d y$

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

Evaluate the integral. $\int \frac{\cos (\ln 18 t)}{t} d t$

Evaluate the integral if it is convergent. $\int_{0}^{\infty} e^{-3 x} d x$

Use long division to evaluate the integral. $\int \frac{x^{2}}{x+3} d x$

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

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

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