   Chapter 7.1, Problem 51E

Chapter
Section
Textbook Problem

# Use integration by parts to prove the reduction formula. ∫ ( ln x ) n d x = x ( ln x ) n − n ∫ ( ln x ) n − 1 d x

To determine

To prove: The given reduction formula using the technique of integration by parts.

Explanation

Proof:

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 reduction formula, (lnx)ndx=x(lnx)nn(lnx)n1dx.

Calculation:

Use integration by parts to solve the integration on left hand side of the formula. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=(lnx)n      dv=dx

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

du=n(lnx)

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