Evaluate the definite integral.

$32$xex$2$ dx$0$

Step $1$

We are asked to evaluate the definite integral

$30$

$2$xex$2$ dx$.$

We first find the corresponding indefinite integral

$2$xex$2$ dx$.$

In this situation, finding the indefinite integral is most easily achieved using the method of integration by substitution.

The first step in this method is to let

u $=$ g$($x$),$

where

g$($x$)$

is part of the integrand and is usually the "inside function" of a composite function

f$($g$($x$)).$

For this indefinite integral, observe that the integrand involves the composite function

ex$2$

with the "inside function"

g$($x$)=$ x$2.$

Therefore, we will choose u $=$

$$x$2$

Awesome job!.

Step $2$

The next step is to find

du $=$ g$\text{'}($x$)$ dx$.$

First find

g$\text{'}($x$).$

g$($x$)=$x$2$

g$\text{'}($x$)=$

$$$2$x

Amazing job.

Therefore, we have the following.

du $=$

$$$2$x

$$dx

Step $3$

We now use the substitution

u $=$ x$2$

and du $=2$x dx to convert the entire integral into one involving only u$.$

$2$xex$2$ dx

$=$

$($ex$2)(2$x dx$)$

$$

$=$

$$eu

$$du