MathematicsMedium64×since 2004Q4037For x∈(−π2,π2)x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)x∈(−2π,2π), if y(x)=∫cosecx+sinxcosecxsecx+tanxsin2xdxy(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d xy(x)=∫cosecxsecx+tanxsin2xcosecx+sinxdx, and \lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0 then y(π4)y\left(\frac{\pi}{4}\right)y(4π) is equal toA−12tan−1(12)-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)−21tan−1(21)Btan−1(12)\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)tan−1(21)C12tan−1(12)\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)21tan−1(21)D12tan−1(−12)\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)21tan−1(−21)Check answerSkip