# Integrate (2x^2+1)e^x^2dx ( Wow, seriously?)

## Homework Statement

Integrate: (2x^2+1)e^x^2dx

## The Attempt at a Solution

I don't even know where to start , I either got to do this by basic substitution or by parts. Basic substitution doesn't help for obvious reasons, so I thought I'd do it by parts, but that would get completely messed up based upon the e^x^2 not having any anti-derivative, so I'd need basic substitution to get rid of it.

I thought of making:

u = e^x^2
ln|u| = x^2
√ln|u| = x
du = 2xe^x^2 dx

Therefore, my equation would be:

(2ln|u| + 1)/(2√[ln|u|]) du

Then I thought about breaking that up:

integrate: (2ln|u|)/(2√[ln|u|]) + 1/(2√[ln|u|])

Simplified to...

(ln|u|)/(√[ln|u|]) + 1/(2√[ln|u|])

Annnndd... now I'm stuck again because that square root is basically the most evilest thing on the entire planet. It limits partial fractions, I can't exactly do trig substitution, and the fraction kills by parts.

Expand out the brackets of the integrand to get two terms, then try to do by parts on the first term only. Hint: $$dv = 2x e^{x^2}dx$$.