   Chapter 8.3, Problem 22E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21–34, find theindefinite integral. ∫ tan 6 3 x d x

To determine

To calculate: The value of the following indefinite integral tan63xdx

Explanation

Given: The provided expression is tan63xdx

Formula Used:

The following trigonometric identities,

tan23x=sec23x1

The following differential formulas,

ddxtanx=sec2x.

ddxxn=nxn1

The following integral formulas,

undu=un+1n+1+C.

(sec2u)du=tanu+C.

dx=x+C

Chain rule of differentiation, ddxf(g(x))=f(x)ddxg(x)

Calculation:

Consider the provided integral;

tan63xdx.

tan63xdx=tan23xtan43xdx=(sec23x1)tan43xdx=(sec23xtan43xtan43x)dx=(sec23xtan43x)dxtan43xdx

Now, apply the linearity property of integration, we have

tan63xdx=(sec23xtan43x)dxtan23xtan23xdx=(sec23xtan43x)dx(sec23x1)tan23xdx=(sec23xtan43x)dx(sec23xtan23xtan23x)dx=(sec23xtan43x)dx(tan23xsec23x)dx+tan23xdx

Now use the following trigonometric identity, tan23x=sec23x1, we have

tan63xdx=(sec23xtan43x)dx(tan23xsec

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