Question

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

Answer 1

By parts:

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