How do you integrate int x^2e^(x^3)x2ex3 by parts?

1 Answer
Dec 18, 2016

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

Explanation:

intx^2e^(x^3) dxx2ex3dx

Let u = x^3u=x3. This makes du = 3x^2 dxdu=3x2dx.

The integral becomes

1/3 int e^(x^3) (3x^2dx) = 1/3 int e^u du13ex3(3x2dx)=13eudu

= 1/3 e^u + C=13eu+C

= 1/3 e^(x^3) + C=13ex3+C

If I am told that I must use parts ,

I'll let u = 1u=1 and dv = x^2e^(x^3) dxdv=x2ex3dx

so that du = 0 dxdu=0dx and v = 1/3e^(x^3)v=13ex3.

And

uv=int v du = 1 * 1/3e^(x^3) - int 1/3e^(x^3) * 0 duuv=vdu=113ex313ex30du

= 1/3 e^(x^3) + C=13ex3+C.