   Chapter 7.1, Problem 46E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0 ). ∫ x 2 sin 2 x   d x

To determine

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

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts in terms of u and v is given by

udv=uvvdu

Given:

The integral, x2sin2xdx.

Calculation:

Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=x2      dv=sin2xdx

Then, the differentiation of u and antiderivative of dv will be

du=2xdx     v=12cos2x

Using the formula above, the given integration will become

x2sin2xdx=x2(12cos2x)(12cos2x)2xdx=x22cos2x+xcos2xdx …… (1)

Integrand in the last term is also product of two functions, so apply integration by parts

xcos2xdx using the variables

u=x     dv=cos2xdx

Then

du=dx      v=12sin2x

Then, the inte

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