   Chapter 8.3, Problem 25E

Chapter
Section
Textbook Problem

Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21–34, find theindefinite integral. ∫ tan 5 x 2 d x

To determine

To calculate: The value of the following indefinite integral tan5x2dx

Explanation

Given: The provided indefinite integral is tan5x2dx

Formula Used:

The following trigonometric identity,

tan2x2=sec2x21

The following concepts based on differentiation,

ddxtanx2=12sec2x2.

ddxcosx2=12sinx2

Calculation:

Consider the provided integral expression tan5x2dx.

tan5x2dx=tan2x2tan3x2dx=(sec2x21)tan3x2dx=(sec2x2tan3x2tan3x2)dx=(sec2x2tan3x2)dx(tan2x2tanx2)dx

Simplify further,

tan5x2dx=(sec2x2tan3x2)dx(sec2x21)tanx2dx=(sec2x2tan3x2)dx(sec2x2tanx2tanx2)dx tan5x2dx=(sec2x2tan3x2)dx(sec2x2tanx2)dx+(tanx2)dx ------(1)

Now, integrate this equation by the method of substitution,

Let consider tanx2=y ------(2)

Now differentiate equation (2) with respect to x, to get

ddxtanx2=dydx

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