Free Printable Integration by Substitution (USub)

1.

Use the indicated substitution to evaluate the integral.

$\int2x\cos(x^2)\ dx\ ,\ \ \ \ u=x^2$

a)

$\sin\left(x^2\right)+C$

b)

$2\sin\left(x^2\right)+C$

c)

$\frac{1}{2}\sin\left(x^2\right)+C$

d)

$4\cos\left(x^2\right)+C$

2.

Use the indicated substitution to evaluate the integral.

$\int x\left(x^2+5\right)^7dx\ ,\ \ \ \ u\ =\ x^2+5$

a)

$\frac{\left(2x+5\right)^8}{8}+c$

b)

$\frac{\left(x^2+5\right)^8}{16}+c$

c)

$\frac{\left(x^2+5\right)^6}{6}+c$

d)

$\frac{\left(2x+5\right)^8}{16}+c$

3.

Use the indicated substitution to evaluate the integral.

$\int\frac{2x}{\sqrt{3+x^2}}dx\ ,\ \ \ u=3+x^2$

a)

$\frac{1}{2\sqrt{3+x^2}}+C$

b)

$\sqrt{3+x^2}+C$

c)

$2\sqrt{3+x^2}+C$

d)

$\frac{2}{\sqrt{3+x^3}}+C$

4.

Use the indicated substitution to evaluate the integral.

$\int-3\cos x\sin\left(\sin x\right)\ dx\ ,\ \ \ u=\sin x$

a)

$3\sec\left(\sin x\right)+C$

b)

$3\sin\left(\sin x\right)+C$

c)

$3\cot\left(\sin x\right)+C$

d)

$3\cos\left(\sin x\right)+C$

5.

Use substitution to evaluate the integral $\int\frac{2x}{\left(x^2+3\right)^3}\ dx$

a)

$-\frac{1}{2\left(x^2+3\right)^2}+C$

b)

$-\frac{3}{2\left(x^2+3\right)^2}+C$

c)

$-\frac{5}{2\left(x^2+3\right)^2}+C$

d)

$-\frac{1}{\left(x^2+3\right)^4}+C$

6.

Use substitution to evaluate the integral $\int\left(3x^3+5\right)^5\cdot27x^2dx$

a)

$\frac{1}{2}\left(3x^3+5\right)^6+C$

b)

$\frac{2}{5}\left(3x^3+5\right)^5+C$

c)

$\frac{5}{6}\left(3x^3+5\right)^6+C$

d)

$\frac{3}{5}\left(3x^3+5\right)^5+C$

7.

Use substitution to evaluate the integral $\int4\sec^24x\ \sqrt{\tan4x}\ dx$

a)

$2\left(\tan4x\right)^{\frac{3}{2}}+C$

b)

$\frac{15}{4}\cdot\left(\tan4x\right)^{\frac{4}{3}}+C$

c)

$\frac{4}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

d)

$\frac{2}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

8.

Use substitution to evaluate the integral

$\int\frac{\left(\ln^6x\right)}{x}\ dx$

a)

$7\left(\ln x\right)^7+C$

b)

$\frac{\ln^6x}{x^2}+C$

c)

$\frac{1}{7}\left(\ln x\right)^7+C$

d)

$\frac{6}{7}\left(\ln x\right)^7+C$

9.

Use substitution to evaluate the integral

$\int\frac{6\cos t}{\left(2+\sin t\right)^2}dt$

a)

$\frac{6}{2+\sin t}+C$

b)

$\frac{3}{2+\sin t}+C$

c)

$-\frac{6}{2+\sin t}+C$

d)

$-\frac{3}{2+\sin t}+C$

10.

Use substitution to evaluate the integral

$\int\ \frac{dx}{x\ln x}$

a)

$\ln\left(\ln x\right)+C$

b)

$\left(\ln x\right)^2+C$

c)

$\frac{1}{x^2}+C$

d)

$\frac{\ln x}{x^2}+C$

11.

Use substitution to evaluate the integral

$\int\ \tan^2x\ \sec^2x\ dx$

a)

$\frac{1}{3}\sec^3x+C$

b)

$\frac{1}{3}\tan^3x+C$

c)

$\frac{2}{3}\tan^{\frac{2}{3}}x+C$

d)

$\frac{1}{3}\sec x\ \tan x+C$

12.

Use substitution to evaluate the integral

$\int\ x^2\sqrt{5+2x^3}\ dx$

a)

$\frac{1}{9}\left(5+2x^3\right)^{\frac{3}{2}}+C$

b)

$\frac{2}{3}\left(5+2x^3\right)^{\frac{3}{2}}+C$

c)

$6x\left(5+2x^3\right)^{\frac{3}{2}}+C$

d)

$\frac{1}{9}x^3\left(5+2x^3\right)^{\frac{3}{2}}+C$

13.

What is the choice of u for ∫ e^x(1+e^x)^(1/2) dx?

a)

e^x

b)

1+e^x

c)

(1+e^x)^(1/2)

d)

(e^x)^(1/2)

14.

Evaluate without a calculator. Show all steps on a sheet of paper.

a)

A

b)

C

c)

D

d)

E

15.

Evaluate without a calculator. Show all steps on a sheet of paper.

a)

A

b)

B

c)

C

d)

D

16.

Evaluate without a calculator. Show all steps on a sheet of paper.

a)

A

b)

B

c)

C

d)

E

17.

Evaluate without a calculator. Show all steps on a sheet of paper.

a)

A

b)

B

c)

C

d)

E

18.

Evaluate without a calculator. Show all steps on a sheet of paper.

a)

A

b)

B

c)

C

d)

D

Integration by Substitution (USub)

Total questions: 18

Worksheet time: 9mins

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