Been at this forever amount of hours now, looking around online, asking fellow students etc. Couldn't find a thread about this, so here goes:
$\int_{-\infty}^{\infty} e^{\big(\frac{-x^2}{a^2}\big)}x^2 \,dx\space\space\space\space\space$ (1)
DI-method table setup gives me:
\begin{array}{|sign|D|I|} \hline & D & I\\ \hline + & x^2 & e^{\big(\frac{-x^2}{a^2}\big)} \\ \hline - & 2x & a^2\sqrt{\pi} \\ \hline + & 2 & xa^2\sqrt{\pi} \\ \hline - & 0 & \frac{1}{2}x^2a^2\sqrt{\pi} \\ \hline \end{array}
This would, if I'm doing the method correctly, allow me to get rid of the integral altogether (because of the $0$ in the D-column at the bottom):
$x^2 \cdot a^2\sqrt{\pi} -x^2a^2\sqrt{\pi} + 2 \cdot \frac{1}{2}x^2a^2\sqrt{\pi} = 0 \space\space\space\space\space (2)$
So, that would make the integral I'm looking to solve: $\int_{-\infty}^{\infty} e^{\big(\frac{-x^2}{a^2}\big)}x^2 \,dx = 0$, which it's not supposed to be.
I have a solution given with partial integration, which I'm not interested in, as I'm doubling down on the DI-method. The answer is supposed to be $\frac{a^2}{2}$. Surely, this must be solvable with the DI-method as well as with the partial integration formula?
Where do I go wrong?