   Chapter 7.3, Problem 8E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ d t t 2 t 2 − 16

To determine

To evaluate: The given integral dtt2t216.

Explanation

Integration involving terms of the form x2a2 can be simplified by using the trigonometric substitution x=asecθ.

Formula used:

The identity, tan2x=sec2x1

Given:

The integral, dtt2t216

Calculation:

Substitute for t as t=4secθ. Take the derivative of the substitution term:

t=4secθdt=4secθtanθdθ

Here, 0θ<π2

Substitute for t and dt in the given integral to get:

dtt2t216=4secθtanθdθ16sec2θ16sec2θ16=116tanθdθsecθsec2θ1

Use the identity tan2x=sec2x1:

dtt2t216=116tanθdθsecθsec2θ1=116

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