Am i doing something wrong? I cant seem to spot my mistake, the answer is in red (given by my teacher).
$\displaystyle \int x^3 \cdot e^{\frac{x^2}{2}} \, dx$
$\displaystyle t = \frac{x^2}{2}$
$\displaystyle dt = x \, dx$
$\displaystyle 2 \int x \cdot \frac{x^2}{2} \cdot e^{\frac{x^2}{2}} \, dx$
substitute ...
$\displaystyle 2 \int t \cdot e^{t} \, dt$
now do parts
While, in differentiating something like $\displaystyle e^{x^2}$ you can just multiply by the derivative of $\displaystyle x^2$, to get $\displaystyle 2xe^{x^2}$, when integrating you cannot just divide by the derivative- it must already be in the integral.