What exactly is it?

Calculus Level 4

For any real number x, let \([x]\) denote the largest integer less than or equal to \(x\) let \(f\) be a real valued function defined on the interval \([-10,10]\) by

\[ \begin{eqnarray} f(x) &=& x - [x], \text { if } f[x] \text{ is odd} \\ f(x) &=& 1 + [x] - x, \text { if } f[x] \text{ is even} \\ \end{eqnarray} \]

Then find the value of \( \displaystyle \frac {\pi^2}{10} \int_{-10}^{10} \bigg [ f(x) \cos(\pi x) \bigg ] \ dx \) is?


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