   Chapter 7.1, Problem 7E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ ( x 2 + 2 x ) cos 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 in terms of u and v is given by

udv=uvvdu

Given:

The integral, (x2+2x)cosxdx.

Calculation:

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

u=x2+2x     dv=cosxdx

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

du=(2x+2)dx     v=sinx

Substitute for variables in the formula above to get

(x2+2x)cosxdx=(x2+2x)(sinx)(sinx)(2x+2)dx

Integrate the last term

(x2+2x)cosxdx=(x2+2x)(sinx)(2xsinx+2sinx)dx=(x2+2x)(sinx)(2sinx)dx(2xsinx)dx=(x2+2x)(sinx)2sinxdx2xsinxdx

=(x2+2x)(sinx)2(cosx)2C12xsinxdx …… (1)

Notice that the last term is against an integral of product of two functions

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