Express the indefinite integral $\int e^{-x^2}dx$ using function $\Phi(x)$.

Question

Express the indefinite integral $\int e^{-x^2}dx$ using function $\Phi(x)$. $\Phi(x)$ is the following special function:

$$\Phi(x) = \frac12 +\frac{1}{\sqrt{2\pi}}\int_0^x e^{-t^2/2}\,dt$$

Answer 1

Hint. Perfom the change of variable $t=\sqrt{2}\:u$ in the integrand of $Φ$, you get $$ \int_0^x e^{-t^2/2}dt=\sqrt{2}\int_0^{x/\sqrt{2}} e^{-u^2}du $$ then you find $$ Φ(x) = 1/2 +(1/\sqrt{π})\int_0^{x/\sqrt{2}} e^{-u^2}dt $$ giving $$ \int_0^x e^{-u^2}du=\sqrt{π}\:Φ(\sqrt{2}\:x)- \sqrt{π}/2. $$ Then observe that $$ \int e^{-u^2}du=\int_0^x e^{-u^2}du+C $$ where $C$ is a constant.