How do you find the antiderivative of #x[e^(x^2)]#?

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Answer 1 · 2015-08-22T16:16:53

Use substitution with #u = x^2#

I see that the question was posted under "Integration by Parts", but we can find this integral using u substitution.

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

Let #u = x^2#, o that #du = 2x dx# and #xdx = 1/2 du#

With this substitution, we get:

#int x e^(x^2) dx = 1/2 int e^u du#

# = 1/2 e^u +C#

# = 1/2e^(x^2) +C#

Check the answer by differentiating.

Note The integral #int xe^(5x)dx# does call for integration by parts, because the substitution #u = 5x# won't get us something we can integrate.

Note 2 I have assumed here that I can use "integral" in place of "antiderivative" without loss of understanding.