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Question Number 165955 by cortano1 last updated on 10/Feb/22

$$\mathrm{C}={\int }_0^{\pi }\frac{\mathrm{dx}}{2+\cos 2\mathrm{x}}=?$$

Answered by MJS_new last updated on 10/Feb/22

$$\underset{0}{\stackrel{\pi }{\int }}\frac{dx}{2+\cos 2x}=2\underset{0}{\stackrel{\pi /2}{\int }}\frac{dx}{2+\cos 2x}$$$$\int \frac{dx}{2+\cos 2x}=$$$$[t=\tan x\to dx=\frac{dt}{t^2+1}]$$$$=\int \frac{dt}{t^2+3}=\frac{\sqrt{3}}{3}\arctan \frac{\sqrt{3}t}{3}=\frac{\sqrt{3}}{3}\arctan \frac{\sqrt{3}\tan x}{3}+C$$$$\Rightarrow$$$$\underset{0}{\stackrel{\pi }{\int }}\frac{dx}{2+\cos 2x}=\frac{\sqrt{3}\pi }{3}$$

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