   Chapter 7.3, Problem 3E

Chapter
Section
Textbook Problem

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

To determine

To evaluate: The integral x24xdx using the given trigonometric substitution.

Explanation

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

Formula used:

The identity, sec2xtan2x=1

Given:

The integral, x24xdx

Substituting term, x=2secθ

Calculation:

Take the derivative of the given substitution term:

x=2secθdx=2secθtanθdθ

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

x24xdx=(2secθ)242secθ2secθtanθdθ=4sec2θ42secθ2secθtanθdθ=2sec2θ1tanθdθ

Use the identity sec2xtan2x=1:

x24xdx=2tan2θtanθdθ=2tanθtanθdθ=2tan2θdθ=2(sec

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