Describe the method used to integrate sinm x cosn x, for m even and n odd.

Answered: Describe the method used to integrate…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.4: Multiple-angle Formulas

Problem 41E

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Question

Describe the method used to integrate sin^m x cosⁿ x, for m even and n odd.

Expert Solution

Step 1

Method to integrate $\sin^{m} x \cos^{n} x$ ,for $m$ even and $n$ odd.

Let $m = 2 k$ and $n = 2 k + 1$

$\begin{array}{rcl} \int \sin^{m} x \cos^{n} x d x & = & \int \sin^{2 k} x \cos^{2 k + 1} x d x \\ = & \int \sin^{2 k} x \cos^{2 k} x \cos x d x \\ = & \int \sin^{2 k} x {(1 - \sin^{2} x)}^{k} \cos x d x \end{array}$

Subsitute $t = \sin x$

$\begin{array}{rcl} \frac{d t}{d x} & = & \cos x \\ d t & = & \cos x d x \end{array}$

$\int \sin^{m} x \cos^{n} x d x = \int t^{2 k} {(1 - t^{2})}^{2 k} d t$

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