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Question Number 123020 by mnjuly1970 last updated on 21/Nov/20

$$...advancedcalculus..$$$$$$$$calculate::\mathrm{\varnothing }={\int }_0^{\pi }\frac{\pi }{1-sin\left(x\right)cos\left(x\right)}dx=???$$$$....................$$

Answered by Dwaipayan Shikari last updated on 21/Nov/20

$${\int }_0^{\pi }\frac{\pi }{1-\frac{1}{2}sin2x}dx$$$$=2{\int }_0^{\mathrm{\infty }}\frac{2\pi }{2-\frac{2t}{1+t^2}}.\frac{1}{1+t^2}dtt=tanx$$$$=2{\int }_0^{\mathrm{\infty }}\frac{\pi }{t^2-t+1}dt=2{\int }_0^{\mathrm{\infty }}\frac{\pi }{(t-\frac{1}{2})+{\left(\sqrt{\frac{3}{4}}\right)}^2}={\left[\frac{4\pi }{\sqrt{3}}{tan}^{-1}\frac{2t-1}{\sqrt{3}}\right]}_0^{\mathrm{\infty }}=\frac{2{\pi }^2}{\sqrt{3}}$$

Commented by mnjuly1970 last updated on 21/Nov/20

$$thankyousomuch...$$

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