   Chapter 7.1, Problem 41E

Chapter
Section
Textbook Problem

# First make a substitution and then use integration by parts to evaluate the integral. ∫ x  ln ( 1 + 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, xln(1+x)dx.

Calculation:

Rewrite the integral by making the substitution 1+x=t. The differential of t will then be

dx=dt

Substitute for 1+x and dx in the integral given:

xln(1+x)dx=(t1)lntdt…… (1)

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

u=lnt      dv=(t1)dt

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

du=1tdt     v=t22t

Using the formula above, the given integration will become

(t1)lntdt=lnt(t22t)(t22t)1tdt

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