Chapter 7.1, Problem 24E

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

# Evaluate the integral. ∫ 0 1 ( x 2 + 1 ) e − x d x

To determine

To evaluate: The given integral using the technique of integration by parts.

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts for definite integral is given by

abf(x)g(x)dx=f(x)g(x)]ababg(x)f(x)dx

Given:

The integral, 01(x2+1)exdx.

Calculation:

Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=x2+1      dv=exdx

Then, the differentiation of u and antiderivative of dv will be

du=2xdx      v=ex

Using the formula above, the given integration will become

01(x2+1)exdx=(x2+1)(ex)]0101(ex)(2x)dx=(x2+1)(ex)]01+201xexdx …… (1)

Integrate the last term using integration by parts with the following substitution:

u=x     dv=exdx

Then

du=dx

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