Hi, it helps when you put your equation in Mathematica format I assume that symbol in the exponent is a Pi
and not an n and that L is greater than zero:
In[72]:= Clear["Global`*"]
In[73]:= sol =
Integrate[Exp[-L/(Pi (t - \[Tau]))]/Sqrt[t - \[Tau]], \[Tau]]
Out[73]= -2 E^(-(L/(\[Pi] (t - \[Tau])))) Sqrt[t - \[Tau]] -
2 Sqrt[L] Erf[Sqrt[L]/(Sqrt[\[Pi]] Sqrt[t - \[Tau]])]
In[74]:= l1 =
Limit[sol, \[Tau] -> t, Assumptions -> L > 0, Direction -> 1]
Out[74]= -2 Sqrt[L]
In[75]:= l2 =
Limit[sol, \[Tau] -> 0, Assumptions -> L > 0, Direction -> -1]
Out[75]= -2 E^(-(L/(\[Pi] t))) Sqrt[t] -
2 Sqrt[L] Erf[Sqrt[L]/(Sqrt[\[Pi]] Sqrt[t])]
In[76]:= l1 - l2
Out[76]= -2 Sqrt[L] + 2 E^(-(L/(\[Pi] t))) Sqrt[t] +
2 Sqrt[L] Erf[Sqrt[L]/(Sqrt[\[Pi]] Sqrt[t])]