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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.

( x + 1 ) e x d x

To determine

To calculate: The infinite integral of (x+1)exdx by use the method of integration by parts.

Answer

Solution:

The infinite integral of (x+1)exdx is xex+C.

Explanation

Given Information:

The provided integral is (x+1)exdx.

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.

(x+1)exdx

Observe from the above integrand that the simplest portion of the integrand is x+1.

Now consider u=x+1 and the remaining factors as dv=exdx.

Therefore,

du=dx

And,

v=ex

Apply the integration by parts method.

udv=uvvdu

Substitute x+1 for u, ex for v, dx for du and solve.

(x+1)exdx=(x+1)exexdx=(x+1)exex+C=xex+exex+C=xex+C

Hence, the infinite integral of (x+1)exdx is xex+C.

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