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I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?

t - Integrate[Abs[t - s]^(-1/2)*s, {s, 0, 1}]

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You just need to give Integrate an assumption:

t - Integrate[Abs[t-s]^(-1/2)*s,{s,0,1}, Assumptions->t ∈ Reals] //TeXForm

$t-\left( \begin{array}{cc} \{ & \begin{array}{cc} \frac{2}{3} \left(2 (-t)^{3/2}+2 \sqrt{1-t} t+\sqrt{1-t}\right) & t\leq 0 \\ -\frac{2}{3} \left(-2 t^{3/2}+2 \sqrt{t-1} t+\sqrt{t-1}\right) & t\geq 1 \\ \frac{2}{3} \left(2 t^{3/2}+2 \sqrt{1-t} t+\sqrt{1-t}\right) & \text{True} \\ \end{array} \\ \end{array} \right)$

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  • $\begingroup$ Thank you for your comment. I just have a question, do you have an idea how to find a numerical solution for Integrate[Abs[t - s]^(-1/2)*u(s), {s, 0, 1}] assuming that we dont know the solution u? $\endgroup$
    – Mutaz
    Commented May 19, 2019 at 19:18
  • $\begingroup$ @Mutaz, better to ask a new question (referencing this answer) than request additional help in the comments. $\endgroup$
    – mikado
    Commented May 19, 2019 at 19:30
  • $\begingroup$ Ok, will do. Thank you! $\endgroup$
    – Mutaz
    Commented May 19, 2019 at 19:31

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