   Chapter 7.1, Problem 55E

Chapter
Section
Textbook Problem

# Use Exercise 51 to find ∫ ( ln x ) 3 d x .

To determine

To evaluate: the given integral using reduction formula

Explanation

The reduction formula is obtained using the technique of integration by parts. 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 reduction formula to be used is given below:

(lnx)ndx=x(lnx)nn(lnx)n1dx

Given:

The integral, (lnx)3dx.

Calculation:

Evaluate the given integral by substituting n=3 in the reduction formula:

(lnx)3dx=x(lnx)33(lnx)31dx=x(lnx)33(lnx)2dx …… (1)

Solve the integral (lnx)2dx by again applying the reduction formula with n as 2.

(lnx)2dx=x(lnx)22(lnx)21dx=x(lnx)22(lnx)dx …… (2)

Once again apply the reduction formula to solve the integral lnx

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