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Could anyone give me some light on how to evaluate $$ \int_{0}^{1} \frac{\sqrt{1-t}\sqrt{a+t}}{t^2}\left(1-e^{-t}-2te^{-t} \right)dt$$ where $a$ is a positive real constant.

I tried several different things including contour integral, integration by parts, and expanding the square root $\sqrt{a+t}$ with the hope that the resultant series may eventually add up to some closed form functions. None of these efforts worked. Thank you for your inputs.

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  • $\begingroup$ Do you have any reason to think there is a closed form? Most integrals don't have one. Maybe the best you can do is numerical methods. $\endgroup$ – Gerry Myerson Aug 15 at 2:03
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    $\begingroup$ Also $1-e^{-t}-2te^{-t} = -t+O(t^2)$ so it doesn't converge when $a \ne 0$ $\endgroup$ – reuns Aug 15 at 2:04
  • $\begingroup$ No, I do not have any good reason for a closed form. Of course if there is one that will be great. Thank you. $\endgroup$ – George Aug 15 at 2:07
  • $\begingroup$ @reuns thank you. Oh... $\endgroup$ – George Aug 15 at 2:41
  • $\begingroup$ Even with $a=0$, I am quite skeptical about a possible closed form (even using special functions). By the way, Welcome to the site ! $\endgroup$ – Claude Leibovici Aug 15 at 3:00

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