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How to calculate $$ \int\limits_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}x}{1+\tan^{a}(x)} \;, $$ where $a$ is a constant? Any hint will be appreciated.

littleZ
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1 Answers1

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Let $$I = \int^{\frac{\pi}{2}}_{0}\frac{1}{1+\tan^n (x)}dx\cdots \cdots (1)$$

Replace $\displaystyle x\rightarrow \frac{\pi}{2}-x$

So $$I = \int^{\frac{\pi}{2}}_{0}\frac{1}{1+\tan^n \left(\frac{\pi}{2}-x \right)}dx = \int^{\frac{\pi}{2}}_{0}\frac{1}{1+\cot^{n}(x)}\cdots \cdots (2)$$

So $$2I = \int^{\frac{\pi}{2}}_{0}\frac{1+\tan^{n}(x)}{1+\tan^{n}(x)}dx = \frac{\pi}{2}\Rightarrow I = \frac{\pi}{4}.$$

juantheron
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