Q:

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}]

A:

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

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

