Free Printable Trigonometric Substitution

1.

Evaluate $\int_{ }^{ }x\sqrt{x^2-9}\ dx$

a)

$\frac{\left(\sqrt{x^2-9}\right)^3}{6}+C$

b)

$9\left(\sqrt{x^2-9}\right)^3+C$

c)

$3\left(\sqrt{x^2-9}\right)^3+C$

d)

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

2.

Evaluate $\int_{ }^{ }\ \frac{1}{x^2\sqrt{9-4x^2}}\ dx$

a)

$\frac{-4x}{9\sqrt{9-4x^2}}+C$

b)

$\frac{-\sqrt{9-4x^2}}{18x}+C$

c)

$\frac{-\sqrt{9-4x^2}}{9x}+C$

d)

$\frac{\sqrt{9-4x^2}}{9x}+C$

3.

Evaluate $\int_{ }^{ }\frac{1}{x^3\sqrt{x^2-4}}\ dx$

a)

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

b)

$\frac{1}{16}\sec^{-1}\left(\frac{x}{2}\right)+\frac{\sqrt{x^2-4}}{8x^2}+C$

c)

$\frac{1}{8}\sec^{-1}\left(\frac{x}{2}\right)+\frac{\sqrt{x^2-4}}{4x^2}+C$

d)

$\frac{1}{8}\sec^{-1}\left(\frac{x}{2}\right)+\frac{\sqrt{x^2-4}}{8x^2}+C$

4.

Evaluate $\int_{ }^{ }\frac{1}{x\sqrt{9+x^2}}\ dx$

a)

$-\frac{1}{3}\ln\left|\frac{\sqrt{9+x^2}+3}{x}\right|+C$

b)

$\frac{1}{3}\ln\left|\frac{\sqrt{9+x^2}+3}{x}\right|+C$

c)

$-\frac{1}{3}\ln\left|\frac{x}{\sqrt{9+x^2}}+\frac{x}{3}\right|+C$

d)

$\frac{1}{3}\ln\left|\frac{x}{\sqrt{9+x^2}}+\frac{x}{3}\right|+C$

5.

Evaluate $\int_{ }^{ }\frac{1}{x\sqrt{1-x^2}}\ dx$

a)

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

b)

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

c)

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

d)

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

6.

Evaluate $\int_{ }^{ }\frac{2x^2}{\sqrt{4-9x^2}}\ dx$

a)

$\frac{8}{27}\sin^{-1}\left(\frac{3x}{2}\right)-\frac{2x\sqrt{4-9x^2}}{9}+C$

b)

$\frac{4}{27}\sin^{-1}\left(\frac{3x}{2}\right)-\frac{3x\sqrt{4-9x^2}}{4}+C$

c)

$\sin^{-1}\left(\frac{3x}{2}\right)-\frac{3x\sqrt{4-9x^2}}{4}+C$

d)

$\frac{4}{27}\sin^{-1}\left(\frac{3x}{2}\right)-\frac{x\sqrt{4-9x^2}}{9}+C$

7.

Evaluate $\int_{ }^{ }x^3\sqrt{25-x^2}\ dx$

a)

$\frac{\left(\sqrt{25-x^2}\right)^5}{25}-\frac{2\left(\sqrt{25-x^2}\right)^3}{25}+C$

b)

$-\frac{\left(\sqrt{25-x^2}\right)^5}{5}+\frac{25\left(\sqrt{25-x^2}\right)^3}{3}+C$

c)

$\frac{\left(\sqrt{25-x^2}\right)^5}{5}-\frac{25\left(\sqrt{25-x^2}\right)^3}{3}+C$

d)

$-\frac{\left(\sqrt{25-x^2}\right)^5}{25}+\frac{2\left(\sqrt{25-x^2}\right)^3}{25}+C$

8.

Evaluate $\int\frac{4}{\left(x^2+2\right)^{\frac{3}{2}}}\ dx$

a)

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

b)

$\frac{4\sqrt{2}x}{\sqrt{x^2+2}}+C$

c)

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

d)

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

9.

Evaluate $\int_{ }^{ }\frac{x^3}{\sqrt{2x^2-16}}\ dx$

a)

$16\sqrt{2x^2-16}+\frac{16}{3}\left(\sqrt{2x^2-16}\right)^3+C$

b)

$4\sqrt{2x^2-16}+\frac{1}{12}\left(\sqrt{2x^2-16}\right)^3+C$

c)

$\frac{\sqrt{2x^2-16}}{4}+\frac{\left(\sqrt{2x^2-16}\right)^3}{4}+C$

d)

$\frac{\sqrt{2x^2-16}}{16}+\frac{\left(\sqrt{2x^2-16}\right)^3}{16}+C$

Trigonometric Substitution

Total questions: 9

Worksheet time: 5mins

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