   Chapter 7.3, Problem 1E

Chapter
Section
Textbook Problem

# Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. ∫ d x x 2 4 − x 2       x = 2 sin θ

To determine

To evaluate: The integral dxx24x2 using the given trigonometric substitution.

Explanation

Integration involving terms of the form a2+x2,a2x2 or x2a2 can be simplified by using trigonometric substitutionfor x.

Formula used:

The identity, sin2x+cos2x=1

Given:

The integral, dxx24x2

Substituting term, x=2sinθ

Calculation:

Take the derivative of the given substitution term:

x=2sinθdx=2cosθdθ

Substitute for x and dx in the given integral to get:

dxx24x2=2cosθdθ(2sinθ)244sin2θ=2cosθdx(2sinθ)244sin2θ=14cosθdxsin2θ1sin2θ

Use the identity sin2x+cos2x=1:

14cosθdθsin2θ1sin2θ=14cosθdθsin2θcos2θ=14cosθdθsin2θcosθ

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