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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.

2 x 2 e 2 x d x

To determine

To calculate: The infinite integral of 2x2e2xdx by use the method of integration by parts.

Explanation

Given Information:

The provided integral is 2x2e2xdx.

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.

2x2e2xdx

Observe from the above integrand that the simplest portion of the integrand is 2x2. So, consider, u=2x2 and the remaining factors as dv=e2xdx. Therefore,

du=4xdx

And,

dv=e2xdx

So, integrate above expression as,

dv=e2xdxv=12e2x

Apply the integration by parts method.

udv=uvvdu

Substitute 2x2 for u, 12e2x for v, e2xdx for dv, 4xdx for du and solve

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