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

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Please find below the solution to the asked query:

We haveI=-π2π2sin4xsin4x+cos4x.dxfx=sin4xsin4x+cos4xf-x=sin4-xsin4-x+cos4-xf-x=sin4xsin4x+cos4xf-x=fxHence fx is even function.-aafx.dx=20afx.dx-π2π2sin4xsin4x+cos4x.dx=20π2sin4xsin4x+cos4x.dxI=20π2sin4xsin4x+cos4x.dx....iUse 0afx.dx=0afa-x.dxI=20π2sin4π2-xsin4π2-x+cos4π2-x.dxI=20π2cos4xsin4x+cos4x.dx...iii+ii2I=20π2sin4xsin4x+cos4x.dx+0π2cos4xsin4x+cos4x.dxI=0π2sin4x+cos4xsin4x+cos4x.dxI=0π2 1.dx=x0π2=π2-0=π2-π2π2sin4xsin4x+cos4x.dx=π2

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