   Chapter 7.1, Problem 35E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 1 2 x 4 ( ln x ) 2 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

Given:

The integral, 12x4(lnx)2dx.

Calculation:

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

u=(lnx)2      dv=x4dx

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

du=2lnxxdx     v=x55

Using the formula above, the given integration will become

12x4(lnx)2dx=(lnx)2x55]1212x552lnxxdx=(lnx)2x55]122512x4lnxdx …… (1)

Integration in the last term also involves a product of two function. So, apply integration by parts with the variables as

u=lnx     dv=x4dx

Then

du=dxx      v=x55

Then, the integration 12x4lnxdx will be

12x4lnxdx=lnx(x55)]1212(x55)dxx=lnx(

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