For x (-( )/(2),( )/(2)), if y(x)= x+ x x x+ x ^2xdx, and x (( )/(2))^-limy(x)=0 then y(( …

Question

For $x\in (-\frac{\pi }{2},\frac{\pi }{2})$, if $y(x)=\int \frac{\csc x+\sin x}{\csc x\sec x+	an x{\sin }^{2}x}dx$, and $\underset{x	o {(\frac{\pi }{2})}^{-}}{lim}y(x)=0$ then $y(\frac{\pi }{4})$ is equal to

Accepted Answer

$\begin{gathered} y(x)=\int \frac{(1+{\sin }^{2}x)\cos x}{1+{\sin }^{4}x}dx \ \sin x=t \ =\int \frac{1+{t}^{2}}{{t}^{4}+1}dt=\frac{1}{\sqrt{2}}{	an }^{-1}\frac{(t-\frac{1}{t})}{\sqrt{2}}+C \ x=\frac{\pi }{2},t=1\, 	herefore C=0 \ y(\frac{\pi }{4})=\frac{1}{\sqrt{2}}{	an }^{-1}(-\frac{1}{2}) \end{gathered}$

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