   Chapter 7.1, Problem 32E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 1 2 ( ln x ) 2 x 3   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, 12(lnx)2x3dx.

Calculation:

Rewrite the given integral with the substitution lnx=t. Then, the limit of integration will change to

x1,tln1x2,tln2

The integral may be rewritten by using lnx=t,1xdx=dt:

12(lnx)2x3dx=0ln2t2x2dt since lnx=tx=etx2=e2t

=0ln2t2e2tdt=0ln2t2e2tdt

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

u=t2      dv=e2tdt Then we may rewrite using formula for integration by parts,

udv=uv-vdu

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

du=2tdt     v=12e2t

Using the formula above, the given integration will become

0ln2t2e2tdt=t2(12e2t)]0ln20ln2(12e2t)2tdt=12t2(e2t)]0ln2+0ln2te2tdt …… (1)

The last term here again involves product of two functions

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