# Hot Integral - 22

Calculus Level 5

$\displaystyle \int\limits_{0}^{\pi} t^3\log^8(2\sin t) \, dt$

If the above integral can be expressed as : $\frac{A\pi^C}{B}+\frac{D\pi^F\zeta(G)^H}{E}+I\pi^J\zeta(K)^L + M\pi^N\zeta(O)\zeta(P)+\frac{Q}{U}\pi^R\zeta(S)\zeta(T)$

Evaluate $$A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R+S+T+U$$

Details and Assumptions

• $$A,B,...,S,T,U$$ all are integers.

• $$\gcd(A,B)=\gcd(D,E)=\gcd(Q,U)=1$$

This is a part of Hot Integrals

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