Integration X Axis Formula

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$\displaystyle {\begin{aligned} I_1 &= \int_{-\infty}^{\infty} f(x - \cot x) \: \mathrm{d}x \\ &\stackrel{x \mapsto \frac{\pi}{2} - x}{=} \int_{-\infty}^{\infty} f \left( \frac{\pi}{2} - x - \cot \left( \frac{\pi}{2} - x \right) \right) \: \mathrm{d}x \\ &= \int_{-\infty}^{\infty} f \left( \frac{\pi}{2} - x -\tan x \right) \: \mathrm{d}x \\ &= \int_{-\infty}^{\infty} \frac{\mathrm{d}x}{1+\left( \frac{\pi}{2} - \frac{\pi}{2} + x + \tan x\right)^2} \\ &= \frac{\mathrm{d}x}{1+(x+\tan x)^2} \\ &= I. \end{aligned}}$

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Integration X Axis Formula

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