TITLE: Evaluation of a definite integral.
QUESTION [5 upvotes]: In my real analysis course I was given this exercise:
Calculate $\displaystyle{\int_0^1e^{x^2}dx}$.
What I did was to write $\displaystyle{e^{x^2}=\sum_{n=0}^\infty\dfrac{x^{2n}}{n!}}$ and conclude that $\displaystyle{\int_0^1e^{x^2}dx=\sum_{n=0}^\infty\dfrac{1}{n!(2n+1)}}$. I still don't know if that is correct. My question is:
Is my answer correct? In any case do we know the exact value of  $\displaystyle{\sum_{n=0}^\infty\dfrac{1}{n!(2n+1)}}$? Is there another way calculating this integral?

REPLY [4 votes]: The answer expressed as a summation is correct. The justification of interchanging integral and summation is justified by the fact that the power series expansion of $e^{x^2}$  converges uniformly in $[0,1]$.