What exactly is it?

Calculus Level 5

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

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

Then find the value of π2101010[f(x)cos(πx)] dx \displaystyle \frac {\pi^2}{10} \int_{-10}^{10} \bigg [ f(x) \cos(\pi x) \bigg ] \ dx is?


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