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Pattern – 33 : Definite integration

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Pattern- 33

$I_n=\ \int_{\pi/4}^{\pi/2}\cot^n{x} dx ,\ then\ I_n+I_{n-2}=\ \frac{1}{n-1};n\geq 2$

Proof : Here

$I_n=\ \left[\frac{-\cot^{n-1}{x}}{n-1}\right]^\frac{\pi}{2}-I_{n-2}$

$=\ \frac{-1}{n-1}\left[0-1\right]-I_{n-2}$

$\therefore\ \ I_n+\ I_{n-2}=\ \frac{1}{n-1}\ :n\geq2$