# Beyond Dirichlet

Calculus Level 5

It can be shown that

$\int_0^\infty \frac{\sin^{2m+2}x}{x^4}\,dx \; = \; \frac{\pi^P (m+1)^2}{Q (2m-1) R^m}C_{Sm}$

for all integers $$m \ge 1$$, where $$P,Q,R,S$$ are positive integers with $$Q,R$$ coprime, and $$C_j$$ is the $$j^\text{th}$$ Catalan number $C_j \; = \; \frac{1}{j+1}{2j \choose j} \qquad j \ge 0 \;.$

Give as answer the concatenation $$PQRS$$ of the four integers $$P,Q,R,S$$. For example, if you think the integral is equal to $$\displaystyle\frac{\pi^2 (m+1)^2}{7(2m-1)3^{m}}C_{5m}$$, give the answer 2735.

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