How do you integrate $int x^2e^(x^3)$ by parts?

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Answer 1 · 2016-12-18T01:50:18

Using integration by parts is very artificial for this integral. Substitution is much more reasonable.

$intx^2e^(x^3) dx$

Let $u = x^3$ . This makes $du = 3x^2 dx$ .

The integral becomes

$1/3 int e^(x^3) (3x^2dx) = 1/3 int e^u du$

$= 1/3 e^u + C$

$= 1/3 e^(x^3) + C$

If I am told that I must use parts ,

I'll let $u = 1$ and $dv = x^2e^(x^3) dx$

so that $du = 0 dx$ and $v = 1/3e^(x^3)$ .

And

$uv=int v du = 1 * 1/3e^(x^3) - int 1/3e^(x^3) * 0 du$

$= 1/3 e^(x^3) + C$ .

How do you integrate int x^2e^(x^3)int x^2e^(x^3) by parts?