Question

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Answer 1

By parts:

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