Integrate $e^{x^2} dx$ I tried $u$-substiution with $u = x^2$ but that leaves me with $x$ values and not a simpler function to integrate.Is there a better way to integrate? 

Represent $e^{x^2}$ with the Maclauren series, then separate the term $\frac{1}{k!}$. You should now have an equation that can be easily integrated using the power rule. Integrate it and multiply the $\frac{1}{k!}$ term back in. We could stop here, but if you notice it equates to something close to the error function, then you can say that it is equal to the error function times other missing terms.
$$e^{x^2}=\sum_{k=0}^\infty\frac{(x^2)^k}{k!}\\
\int e^{x^2}=\sum_{k=0}^\infty\frac{1}{k!}\int x^{2k}dx\\
\int e^{x^2}=\sum_{k=0}^\infty\frac{x^{2k+1}}{k!(2k+1)}\\
\sum_{k=0}^\infty\frac{x^{2k+1}}{k!(2k+1)}=\int_0^x e^{-t^2}dt\\
erf(x)=\frac{2}{\sqrt\pi}\int_0^x e^{t^2}dt\\
\int e^{x^2}=-\frac{i\sqrt\pi}{2}erf(ix)$$
