How to calculate $\int e^{\frac{x^2}{2}} \cdot x^{3} dx$? $\int e^{\frac{x^2}{2}} \cdot x^{3} dx=?$
I tried to do the substitution $du = x^3dx$, so that $u=\frac{x^4}{4}$. Then,
$\int e^{\frac{x^2}{2}} \cdot x^{3} dx= \int e^{\sqrt{u}} du$
However, I would have to do a second substitution ($v = \sqrt{u}$) which would give a product again.
How should I proceed?

Let $t=\frac{x^2}{2}, dt=xdx$ to get 
$\displaystyle \int e^{\frac{x^2}{2}}x^3\;dx=\int e^t(2t)dt=2\int te^t\;dt=2\left[te^t-e^t\right]+C=2\left[\frac{x^2}{2}e^{\frac{x^2}{2}}-e^{\frac{x^2}{2}}\right]+C$
$\;\;\;\;=\displaystyle e^{\frac{x^2}{2}}\left[x^2-2\right]+C$
