Hot Integral-3

Calculus Level 5

\[ \large \int_0^\infty \frac{t^2 e^{-3t} }{2-e^{-t}} \, dt \]

The above integral can be expressed as

\[\displaystyle A\zeta(B) - \frac{C}{D} + \frac{E}{F}\log^G(H) - \frac{J}{K}\pi^L\log(M)\]

for positive integers \(A,B,C,D,E,F,G,H,J,K,L \) and \(M \). And \( \gcd(C,D)=\gcd(E,F)=\gcd(J,K)=1\) with \(H\) and \(M\) are square free.

Evaluate \(A+B+C+D+E+F+G+H+J+K+L+M\).

\(\blacksquare\) This is a part of Hot Integrals

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