What is the value this integral ?

Surprisingly it's always zero when $n \not = m$, but when $n = m$ the value is exactly $\pi$!

How so ? Suppose $n \not = m$, similar steps used to evalue $\displaystyle \int_{-\pi}^{\pi}\sin{nx}\cos{mx}\,dx$ can be used to integrate and show that the value is zero. When $n = m$, however, some of the terms "collapse" early into the constant $\displaystyle \frac{1}{2}$ before the integration, and that's the key step to show why the value ends up as $\pi$. Rather cool.

# posted by rot13(Unafba Pune) @ 5:53 AM