Hot Integral - 24

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

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.

• $k>0 , p \in R$

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This is a part of Hot Integrals .