Jaya Legge

2021-10-04

Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions
Use integration by parts to integrate functions.
Integrate products of powers of trigonometric functions.
Evaluate
$\int {\mathrm{csc}}^{3}x{\mathrm{cot}}^{5}xdx$

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Step1
Given the integral
$\int {\mathrm{csc}}^{3}x{\mathrm{cot}}^{5}xdx$
To Evaluate the integral
$I=\int {\mathrm{csc}}^{3}x{\mathrm{cot}}^{5}xdx$
$=\int -\mathrm{csc}x\mathrm{cot}x\left\{\left(-{\mathrm{csc}}^{2}x\left\{\left({\mathrm{cot}}^{2}x\right\}\right)\right\}\right)dx$
$=\int -\mathrm{csc}x\mathrm{cot}x\left\{\left(-{\mathrm{csc}}^{2}x{\left\{\left({\mathrm{csc}}^{2}x-1\right\}\right)}^{2}\right\}\right)dx$
Step 2
Substitute
$u=\mathrm{csc}x$
$du=-\mathrm{csc}x\mathrm{cot}xdx$
$I=-\int {u}^{2}{\left\{\left({u}^{2}-1\right\}\right)}^{2}du$
$I=-\int \left\{\left({u}^{6}-2{u}^{4}+{u}^{2}\right\}\right)du$
$=-\left\{\left[\frac{{u}^{7}}{7}-2\left\{\left(\frac{{u}^{5}}{5}\right\}\right)+\frac{{u}^{3}}{3}\right\}\right]+c$
$=-\frac{{\mathrm{csc}}^{7}x}{7}+\frac{2{\mathrm{csc}}^{5}x}{5}-\frac{{\mathrm{csc}}^{3}x}{3}+c$

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