   Chapter 7.2, Problem 30E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 π / 4 tan 4 t   d t

To determine

To evaluate: The trigonometric integral 0π4tan4tdt

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 secant in the integral is even, save one sec2x factor and use the identity sec2x=1+tan2x to rewrite other terms in tangent function form:

tanmxsec2kxdx=tanmx(1+tan2x)k1sec2xdx

Then, use the substitution u=tanx

Given:

The integral, 0π4tan4tdt.

Calculation:

Rewrite the given integral in terms of square of tangent:

0π4tan4tdt=0π4tan2ttan2tdt

Use the identity tan2x=sec2x1 to substitute for one of the square of tangent term:

0π4tan2ttan2tdt=0π4(sec2t1)tan2tdt=0π4sec2ttan2tdt0π4tan2tdt …… (1)

Solve each integral one by one. First, consider 0π4sec2ttan2tdt. Use the substitution u=tant and du=sec2tdt

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