How do you integrate #int x^2 e^(x^2 ) dx # using integration by parts?

1 Answer
Jim H
Apr 30, 2016

See the explanation section below.

Explanation:

To integrate #x# to a power times #e# to a power, we expect to differentiate the #x# and integrate the #e# to a power

#int x^2 e^(x^2 ) dx#

In order to integrate #e^(x^2) dx# we need an #x# so that we can use substitution.

#int x^2 e^(x^2) dx = int x e^(x^2)x dx# .

Let #u = x# and #dv = e^(x^2)x dx#

The #du = 1 dx# and #v = 1/2 e^(x^2)#

#int x^2 e^(x^2) dx = 1/2xe^(x^2) - 1/2 int e^(x^2) dx#.

Now we need to stop.

#int e^(x^2) dx# has no closed form solution using elementary functions. The integral has a name and some series approximations, but that's the best we can do.

You can read more about it here at Wolfram and here at Wikipedia

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