   Chapter 6, Problem 9RE ### 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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# Evaluating a Definite Integral In Exercises 9–12, use integration by parts to evaluate the definite integral. ∫ 1 e 6 x   ln   x   d x

To determine

To calculate: The definite integral of 1e6xlnxdx by using the method of integration by parts.

Explanation

Given Information:

The provided integral is 1e6xlnxdx.

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.

1e6xlnxdx

In the above integrand, the simplest portion of the integrand is lnx. So, consider, u=lnx and the remaining factors as dv=6xdx. Therefore,

du=1xdx

And,

dv=6xdx

Integrate the above expression for v.

dv=6xdxv=3x2

Again, apply the integration by parts.

udv=uvvdu

Substitute lnx for u, 3x2 for v, 6xdx for dv and 1xdx for du

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