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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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.

ln x x d x

To determine

To calculate: The infinite integral of lnxxdx by use the method of integration by parts.

Explanation

Given Information:

The provided integral is lnxxdx.

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,

lnxxdx

Observe from the above integrand that the simplest portion of the integrand is lnx. So, consider, u=lnx and the remaining factors as dv=1xdx.

Therefore,

du=1xdx

And,

dv=1xdx

So, integrate above expression as,

dv=1xdxv=2x

Apply the integration by parts method

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