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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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Evaluating a Definite integral In Exercises 9–12, use integration by parts to evaluate the definite integral.

0 1 x e x / 4 d x

To determine

To calculate: The definite integral of 01xex/4dx by using the method of integration by parts.

Explanation

Given Information:

The provided integral is 01xex/4dx.

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,

01xex/4dx

Also, the above integrand is written as,

01x(ex/4)dx

In the above integrand, the simplest portion of the integrand is x. So, consider, u=x and the remaining factors as dv=ex/4dx.

Therefore,

du=dx

And,

dv=ex/4dx

Integrate the above expression for v.

dv=ex/4dxv=4ex/4

Again, apply the integration by parts.

udv=uvvdu

Substitute x for u, 4ex/4 for v, ex/4dx for dv and dx for du

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