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.

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