   Chapter 7.2, Problem 54E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0 ). ∫ sec 4 ( 1 2 x )   d x

To determine

To evaluate: The given integral sec4(12x)dx and verify by plotting both the function and its anti derivative.

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, sec4(12x)dx.

Calculation:

Rewrite the given integral in terms of tangent, saving one sec2x term:

sec4(12x)dx=sec2(12x)sec2(12x)dx

Use the identity sec2x=1+tan2x

sec2(12x)sec2(12x)dx=(1+tan2(12x))sec2(12x)dx

Use the substitution u=tan(12x) and du=12sec2(12x)

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