[Math] Help in evaluating $\displaystyle\int\ \cos^2\Big(\arctan\big(\sin\left(\text{arccot}(x)\right)\big)\Big)\ \text{d}x$

calculusindefinite-integralsintegrationtrigonometric-integrals

Is there an easy way to prove this result?

$$\int\ \cos^2\Big(\arctan\big(\sin\left( \text{arccot}(x)\right)\big)\Big)\ \text{d}x = x – \frac{1}{\sqrt{2}}\arctan\left(\frac{x}{\sqrt{2}}\right)$$

I tried some substitutions but I got nothing helpful, like:

  • $x = \cot (z)$

I also tried the crazy one:

  • $x = \cot(\arcsin(\tan(\arccos(z))))$

Any hint?

Thank you!

Best Answer

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$\alpha = \cot^{-1} x\\ \beta = \tan^{-1} (\sin \alpha)$

Use trig indentities to find $\csc \alpha$ and $\sin \alpha$

$\sin\alpha = \tan \beta$

Use similar identities to find $\sec \beta$ and $\cos\beta$

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