Integral value of sin^4x/sin^4x+cos^4x dx from limit -pi/2 to pi/2

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{I}={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx} \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}} \\ \Rightarrow \mathrm{f}\left(-\mathrm{x}\right)=\frac{{\sin }^4\left(-\mathrm{x}\right)}{{\sin }^4\left(-\mathrm{x}\right)+{\cos }^4\left(-\mathrm{x}\right)} \\ \Rightarrow \mathrm{f}\left(-\mathrm{x}\right)=\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}} \\ \Rightarrow \mathrm{f}\left(-\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right) \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{even} \mathrm{function}. \\ \Rightarrow {\int }_{-\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{dx}=2{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{dx} \\ \Rightarrow {\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx} \\ \Rightarrow \mathrm{I}=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}....\left(\mathrm{i}\right) \\ \mathrm{Use} {\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{dx}={\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{a}-\mathrm{x}\right).\mathrm{dx} \\ \Rightarrow \mathrm{I}=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\left({\displaystyle \frac{\mathrm{\pi }}{2}}-\mathrm{x}\right)}{{\sin }^4\left(\frac{\mathrm{\pi }}{2}-\mathrm{x}\right)+{\cos }^4\left(\frac{\mathrm{\pi }}{2}-\mathrm{x}\right)}.\mathrm{dx} \\ \Rightarrow \mathrm{I}=2{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}...\left(\mathrm{ii}\right) \\ \left(\mathrm{i}\right)+\left(\mathrm{ii}\right) \\ \Rightarrow 2\mathrm{I}=2\left\{{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}+{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}\right\} \\ \Rightarrow \mathrm{I}=\left\{{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}{{\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}}.\mathrm{dx}\right\} \\ \Rightarrow \mathrm{I}={\int }_0^{\frac{\mathrm{\pi }}{2}} 1.\mathrm{dx}={\left[\mathrm{x}\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ =\frac{\mathrm{\pi }}{2}-0=\frac{\mathrm{\pi }}{2} \\ {\mathbf{\int }}_{\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{2}}}^{\frac{\mathbf{\pi }}{\mathbf{2}}}\frac{{\mathbf{sin}}^{\mathbf{4}}\mathbf{x}}{{\mathbf{sin}}^{\mathbf{4}}\mathbf{x}\mathbf{+}{\mathbf{cos}}^{\mathbf{4}}\mathbf{x}}\mathbf{.}\mathbf{dx}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}} \end{gathered}$$

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