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Pattern – 32 : Definite integration

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Pattern – 32

$I_n=\ \int_{0}^{\pi/4}\tan^n{x} dx$ then $I_n+I_{n-2}=\ \frac{1}{n-1};n\geq 2$

Proof :

From pattern 27 of indetinite integration

$I_n=\ \left[\frac{\tan^{n-1}{x}}{n-1}\right]_0^{\frac{\pi}{4}}-I_{n-2}$

$\therefore\ \ I_n+I_{n-2}=\ \frac{1}{n-1}\left[\left(1\right)^{n-1}-0\right]$

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