   Chapter 6, Problem 4RE ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Integration by Parts In Exercises 1–8, use integration by parts to find the indefinite integral. ∫ x   ln   4 x   d x

To determine

To calculate: The infinite integral of xln4xdx by using the method of integration by parts.

Explanation

Given Information:

The provided integral is xln4xdx.

Formula used:

The method of integration by parts:

If v and u are two differentiable function of x. Then,

udv=uvvdu

Steps to solve the integral problems:

Step1: At first find the most complicated portion of the integrand and try to letter it as dv so that it can fit a fundamental integration rule. Then, the remaining factor or factors of the integrand will be u.

Step2: First find the factor whose derivative is simple and consider it as u and then the remaining factor or factors of the integrand will be dv and dv should always include the term dx of the original integrand.

Calculation:

Recall the provided integral.

xln4xdx

Observe from the above integrand that the simplest portion of the integrand is ln4x. So, consider, u=ln4x and the remaining factors as dv=xdx.

Therefore,

du=14x(4)dx=1xdx

And,

dv=xdx

So, integrate above expression as,

dv=xdxv=12x2

Apply the integration by parts method

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