   Chapter 7.2, Problem 42E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ sin 2 θ   sin 6 θ   d θ

To determine

To evaluate: The trigonometric integral sin2θsin6θdθ

Explanation

Trigonometric integral of the form sinmxsinnxdx can be solved using the identity for sinAsinB.

Formula used:

The identity sinAsinB=12[cos(AB)cos(A+B)].

Given:

The integral, sin2θsin6θdθ.

Calculation:

Use the identity sinAsinB=12[cos(AB)cos(A+B)] with A as 2θ and B as 6θ:

sin2θsin6θdθ=12(cos(2θ6θ)cos(2θ+6θ))dθ=12(cos(4θ)cos8θ)dθ=12(cos4θcos8θ)dθ

Integrate to get:

12(cos4θcos8θ)dθ=12

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