Winigefx

2022-06-24

Find the interval for which $2\mathrm{arctan}x+\mathrm{arcsin}\frac{2x}{1+{x}^{2}}$ is an independent of x?

alisonhleel3

Expert

HINT:
Let $\mathrm{arctan}x=y\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}-\frac{\pi }{2}\le y\le \frac{\pi }{2}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}-\pi \le 2y\le \pi$
$\mathrm{arcsin}\frac{2x}{1+{x}^{2}}=\mathrm{arcsin}\left(\mathrm{sin}2y\right)$

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