   Chapter 7.2, Problem 51E

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 ). ∫ x sin 2 ( x 2 ) d x

To determine

To evaluate: The given integral and verify by plotting both the function and its antiderivative

Explanation

Trigonometric integral can be solved using the various trigonometric identities which simplifies the integrand.

Formula used:

The identity cos2x=12sin2x

Given:

The integral, xsin2(x2)dx.

Calculation:

Rewrite the given integral using the substitution t=x2, then dt=2xdx.

xsin2(x2)dx=sin2(t)dt2=12sin2(t)dt

Use the identity cos2x=12sin2x to reduce the power of sine:

12sin2(t)dt=12(1cos2t)2dt=14(

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