Determine monotone intervals of a function Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows.
$$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' - \left(\int_1^{x^2} te^{-t^2}dt\right)' \\
= 2x \int_1^{x^2} e^{-t^2}dt  + 2x^3e^{-x^4} - 2x^3e^{-x^4} \\
= 2x \int_1^{x^2} e^{-t^2}dt. $$
But it seems that we are not able to compute $\int_1^{x^2} e^{-t^2}dt$ explicitly. Thank you very much.

Note that $f'(x)=0 \iff x \in \{ -1, 0, 1\} $. It suffices to show that $ \displaystyle\int_{1}^{x^2} e^{-t^2} \, \mathrm{d}t $ is positive for $|x|>1$ and negative for $|x|<1$. Do you see why this is? 
