   Chapter 7.2, Problem 28E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ tan 5 x sec 3 x   d x

To determine

To evaluate: The trigonometric integral tan5xsec3xdx

Explanation

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

Formula used:

When power of tangent in the integral is odd, save one secxtanx factor and use the identity tan2x=sec2x1 to rewrite other terms in secant function form:

tan2k+1xsecnxdx=(sec2x1)ksecn1xsecxtanxdx

Then, use the substitution u=secx

Given:

The integral, tan5xsec3xdx.

Calculation:

Rewrite the given integral in terms of secant, saving one secxtanx term:

tan5xsec3xdx=tan4xsec2xsecxtanxdx

Use the identity tan2x=sec2x1

tan4xsec2xsecxtanxdx=(tan2x)2sec2xsecxtanxdx=(sec2x1)2sec2xsecxtanxdx

Use the substitution u=secx and du=secxtanxdx

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