I have integral: $\int_{-\pi}^{\pi}e^{\frac{-|x|^2}{t}}\theta(t)dx$, where $\theta(t) = 0$ if $t \leq \tau$ and $\theta(t) = 1$ if $t \geq \tau$. I have no idea what I can do with Heaviside function and I tried evaluete integral without it. Nothing succeeded. I asked for help from Wolfram Alpha and Wolfram show me this answer: $\int_{-\pi}^{\pi}e^{\frac{-|x|^2}{t}}dx$ = $\sqrt{\pi} \sqrt{t}$ erf$\big(\frac{\pi}{\sqrt{t}}\big)$, where erf - error function. And I can't understand how..
Thank you in advance.
$$ \begin{align} \int_{-\pi}^{\pi}e^{\frac{-|x|^2}{t}}\theta(t)dx &=\theta(t)\int^\pi_{-\pi}e^{-x^2/t}dx \\ &=\theta(t)\int^\pi_{-\pi}e^{-(x/\sqrt t)^2}dx \\ &=\theta(t)\int^{\pi/\sqrt t}_{-\pi/\sqrt t}e^{-u^2}\sqrt t~du \qquad{(1)}\\ &=\theta(t)\sqrt t\int^{\pi/\sqrt t}_{-\pi/\sqrt t}e^{-u^2}du \\ &=\theta(t)\sqrt t\sqrt \pi\operatorname{erf}\left(\frac{\pi}{\sqrt t}\right) \qquad{(2)}\\ \end{align} $$
$(1)$: let $u=\frac{x}{\sqrt t}$.
$(2)$: due to the definition $\displaystyle{\operatorname{erf}(x):=\frac1{\sqrt\pi}\int^x_{-x}e^{-t^2}dt}$.