   Chapter 7.2, Problem 2E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ sin 3 θ   cos 4 θ   d θ

To determine

To evaluate: The trigonometric integral sin3θcos4θdθ

Explanation

Trigonometric integral of the form sinmxcosnxdx can be solved using strategies depending on whether m and n are odd or even.

Formula used:

When power of sine in the integral is odd, save one sine factor and use the identity sin2x=1cos2x to rewrite other terms in cosine function form:

sin2k+1xcosnxdx=cosnx(1cos2x)ksinxdx

Then, use the substitution u=cosx

Given:

The integral, sin3θcos4θdθ.

Calculation:

Rewrite the given integral in terms of cosine, saving one sine term:

sin3θcos4θdθ=sin2θcos4θsinθdθ

Use the identity sin2x+cos2x=1, to convert remaining terms into cosine:

sin2θcos4θsinθdθ=(1cos2θ)cos4θsinθdθ

Use the substitution u=cosθ and du=sinθdθ

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Calculus: Early Transcendentals 