# Singular Integration

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}]

You just need to give Integrate an assumption:
t - Integrate[Abs[t-s]^(-1/2)*s,{s,0,1}, Assumptions->t ∈ Reals] //TeXForm

$$t-\left( \begin{array}{cc} \{ & \begin{array}{cc} \frac{2}{3} \left(2 (-t)^{3/2}+2 \sqrt{1-t} t+\sqrt{1-t}\right) & t\leq 0 \\ -\frac{2}{3} \left(-2 t^{3/2}+2 \sqrt{t-1} t+\sqrt{t-1}\right) & t\geq 1 \\ \frac{2}{3} \left(2 t^{3/2}+2 \sqrt{1-t} t+\sqrt{1-t}\right) & \text{True} \\ \end{array} \\ \end{array} \right)$$
• 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? – Mutaz May 19 at 19:18