   Chapter 8.3, Problem 26E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral Involving Secant and Tangent In Exercises 19–32, find the indefinite integral. ∫ tan ⁡ 5   2 x   sec ⁡ 4   2 x   d x

To determine

To calculate: The value of integral tan52xsec42xdx.

Explanation

Given:

The provided expression is tan52xsec42xdx.

Formula used:

The trigonometric rule is sec2x=1+tan2x.

The integration rule is tanxdx=sec2x.

Calculation:

Consider the function tan52xsec42xdx.

Break it in two terms,

tan52xsec42xdx=tan52x(sec22x)(sec22x)dx

Recall the trigonometric rule is sec2x=1+tan2x.

tan52xsec42xdx=tan52x(1+tan22x)(sec22x)dx …… (A)

Let 2x=u …… (1)

Differentiate both sides.

2dx=du

Write it as,

dx=12du …… (2)

Substitute values of 2x and sec2xdx from the equation (1) and (2) respectively in the equation (A).

tan52xsec42xdx=12tan5u(1+tan2u)(sec2u)du …… (B)

Let tanu=t …… (iii)

Differentiate both sides

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