# Hot Integral - 24

**Calculus**Level 5

\[\large \displaystyle I(k,p)=\lim_{n \to 0} \frac{1}{\pi}\int\limits_{-\infty}^{-(t-p)}\int\limits_{0}^{\infty}\dfrac{\sin(u/n)\log(p-t)}{ut^{1-k}} \, dt \; du \]

The above expression can be expressed as \[-p^{Ak}k^B[\psi(Ck+D)+p^E\log(\gamma)]\] Evaluate \(I(A+B+C,A+B+C+D+E)\)

**Details and Assumptions**

\(A,B,C,D,E \)are all integers.

\(k>0 , p \in R\)