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

[RoganSarine]

Homework Statement

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

Homework Equations

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.

 

[Mandark]

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.

 

[RoganSarine]

huh... This looks a lot simpler than my way... Let's see if I can finish it :P

 

[RoganSarine]

Huh, wow... Thanks, I didn't even THINK about being able to cancel the integrals that result.