   Chapter 8.3, Problem 27E

Chapter
Section
Textbook Problem

Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21–34, find the indefinite integral. ∫ tan 3 2 t sec 3 2 t   d t

To determine

To calculate: The indefinite integral of the following expression tan32tsec32tdt

Explanation

Given:

The given expression is tan32tsec32tdt

Calculation:

Consider the integral tan32tsec32tdt

By evaluating the following integral.

Rewrite tan32tsec32t as tan22tsec22t(tan2tsec2t)

Now consider the following expression,

tan32tsec32tdt=tan22tsec22t(tan2tsec2t)dt=(sec22t1)sec22t(tan2tsec2t)dt

Let’s assume, sec2t=x (i)

By differentiating both sides,

2sec2ttan2tdt=dxsec2ttan2tdt=dx2 (ii)

By substituting (i) and (ii) in above integral to get,

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