Question

$\mathtt{help\,me}\\\mathtt{my\:universty\:homewok}$
$\mathtt{erf(x)=\dfrac{2}{\sqrt{\pi}}\int\limits^x_0{e^{-t^2}} \, dt}}$
$\mathtt{\int\limits {erf(x)} \, dx=? }$
$\mathtt{Make\:proof!}$
^-^

Answer 1

By parts:

$\int \frac{2}{\sqrt{\pi } }(\int_0^xe^{-t^2}dt) dx = x\frac{2}{\sqrt{\pi } }\int_0^xe^{-t^2}dt - \int \frac{2}{\sqrt{\pi } }xd(\int_0^xe^{-t^2}dt) = \\=x*erf(x) - \frac{2}{\sqrt{\pi } }\int xe^{-x^2}dx = x*erf(x) + \frac{1}{\sqrt{\pi }} \int e^{-x^2}d(-x^2) =\\=xerf(x) + \frac{e^{-x^2}}{\sqrt{\pi} } + C$