M Cauchy Présente

Calculus Level 5

The integral \[ \int_0^\infty e^{\cos x} \sin(\sin x)\,\frac{dx}{x} \] can be expressed as \[ \frac{1}{P}\pi^Q (Re - S) \] where \(P,Q,R,S\) are positive integers where \(P\), \(R\) and \(S\) are pairwise coprime.

Write the answer as the concatenation of \(PQRS\) of the integers \(P,Q,R,S\). For example, if you think that the integral is equal to \(\tfrac13\pi^2(4e - 1)\), give the answer 3241.

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