How to show that $f$ is monotone? 
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by
  $$f(x) = \int_{e^{x^3 +x}}^{1+e^{x^3+x}}e^{r^2} dr$$
  for all $x\in\mathbb{R}$. Prove that $f$ is monotone.

I was thinking about Leibniz rule. Actually  my main problem is that how to show that f is continuous...
I am struck at this problem as I am in fear to solve this problem.
Please help me and remove the fear from my mind.....
Thank you.

For the first derivative of your integral we get $$e^{x^3+e^{2 \left(x^3+x\right)}+x} \left(e^{2
   e^{x^3+x}+1}-1\right) \left(3 x^2+1\right)$$ this derivative is positive.
