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.


Problem Loading...

Note Loading...

Set Loading...