0-2pi-dx-1-sinxcosx-4piln2-3-

Question Number 216695 by sniper237 last updated on 16/Feb/25

\int_{0}^{2 π} \frac{d x}{1 + s i n x c o s x} \overset{?}{=} \frac{4 π l n 2}{\sqrt{3}}

Answered by Ghisom last updated on 16/Feb/25

\underset{0}{\int^{2 π}} \frac{d x}{1 + \sin x \cos x} = 8 \underset{π / 4}{\int^{3 π / 4}} \frac{d x}{2 + \sin 2 x} =

= 8 \underset{0}{\int^{π / 2}} \frac{d x}{2 + \cos 2 x} =

[t = \tan x]

= 8 \underset{0}{\int^{\infty}} \frac{d t}{t^{2} + 3} = \frac{8 \sqrt{3}}{3} {[\arctan \frac{\sqrt{3} t}{3}]}_{0}^{\infty} =

= \frac{4 π \sqrt{3}}{3}

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