Definite Integration Question - None

Question

For a positive integer $n$, let $I_n = \int_{-\pi}^{\pi} \left(\frac{\pi}{2} - |x|\right) \cos(nx) dx$. Find the value of $[I_1 + I_2 + I_3 + I_4]$ where $[\cdot]$ denotes the greatest integer function.

Accepted Answer

Solution: The integrand is even in $x$ and even in $\cos(nx)$. Compute $I_n = 2\int_0^{\pi} \left(\frac{\pi}{2} - x\right) \cos(nx) dx$. Using integration by parts, $I_n = \frac{4}{n^2}$ for odd $n$ and $I_n = 0$ for even $n$. Thus $I_1 + I_2 + I_3 + I_4 = 4 + 0 + \frac{4}{9} + 0 = 4.\overline{4}$, so $[4.\overline{4}] = 4$.

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