show-that-0-x-e-xt-e-t-2-dt-e-x-2-4-x-0-e-t-2-4-

Question Number 210839 by klipto last updated on 19/Aug/24

show that \int_{0}^{x} e^{xt} e^{- t^{2}} dt = e^{\frac{x^{2}}{4}} {\int_{0}}^{x} e^{- \frac{t^{2}}{4}}

Commented by Frix last updated on 20/Aug/24

\underset{0}{\int^{x}} e^{x t} e^{- t^{2}} d t = [u = 2 t - x]

= \frac{e^{\frac{x^{2}}{4}}}{2} \underset{- x}{\int^{x}} e^{- \frac{u^{2}}{4}} d u =

= e^{\frac{x^{2}}{4}} \underset{0}{\int^{x}} e^{- \frac{u^{2}}{4}} d u

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