Integration by Parts - Choice of variables

[takarin]

Homework Statement

I'm getting different results when choosing my u & dv for Integration by Parts on the following integral:

\int 2x^3 e^x^2 dx
(Note, the exponent on 'e' is x^2)

This yields the correct solution:
u = x^2
dv = 2x e^x^2 dx

du = 2xdx
v = e^x^2

However, I have tried using this instead (*)

u = 2x^3
dv = e^x^2 dx

du = 6x^2 dx
v = (e^x^2) / 2x

and this is yielding the incorrect solution (see 3.)

Homework Equations

Integration by Parts:
\int udv = uv - \int vdu

The Attempt at a Solution

The correct solution turns out to be
x^2 e^x^2 - e^x^2 + C

When I use my other choice of variables (*), I get (using IBP)
\int 2x^3 e^x^2 dx = 2x^3 e^x^2 / 2x - \int e^x^2 / 2x * 6x^2 dx
which leads to:
x^2 e^x^2 - 3/2 e^x^2 + C

which is different from the other choice of variables.

I've looked over both choices of variables, and I don't know why the second choice (*) comes up with a different solution.

Thanks for the help!

 

[vela]

Your error is that the integral of e^{x^2} isn't e^{x^2}/(2x). (I'm assuming you meant to have parentheses on the bottom, but perhaps not. Regardless, the answer is incorrect either way.)

 

[takarin]

Ah! Silly me, I've been staring at it and completely overlooked that. Thanks for your quick reply, I'll avoid such carelessness in the future :)