   Chapter 7.1, Problem 23E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 1 / 2 x cos x   π x   d x

To determine

To evaluate: The given integral using the technique of integration by parts.

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 for definite integral is given by

abf(x)g(x)dx=f(x)g(x)]ababg(x)f(x)dx

We may denote f(x) by u and g(x) by dv.

Given:

The integral, 012xcosπxdx.

Calculation:

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

u=x      dv=cosπxdx

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

du=dx      v=1πsinπx

Using the formula above, the given integration will become

012xcosπxdx=x(1πsinπx)]012012(1πsinπx)dx

Integrate the last term:

01

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