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.

The integral $\displaystyle\int_1^{x^2} e^{-t^2}\,dt$ is positive when $x^2>1$ and negative when $0\le x^2<1$.  That means it is positive when $x>1$ or $x<-1$, and negative when $-1<x<1$.  You're multiplying it by $2x$ which is positive when $x>0$ and negative when $x<0$.
So the derivative is negative when $x<-1$, positive when $-1<x<0$, negative when $0<x<1$, and positive when $x>1$.
