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Evaluate : int0^(pi/2)(xsinxcosx)/(sin^4...

Evaluate : `int_0^(pi/2)(xsinxcosx)/(sin^4x+cos^4x)\ dx`

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Given:`int_0^(pi/2)(xsinxcosx)/(sin^4x+cos^4x)\ dx`
`I=int_0^(pi/2)(xsinxcosx)/(sin^4x+cos^4x)\ dx....(i)`
`I=int_0^(pi/2)((pi/2-x)cosxsinx)/(cos^4x+sin^4x)\ dx.....(ii)`
Now add (i) and (ii) we get
`2I=pi/2 int_0^(pi/2)(sinxcosx)/(sin^4x+cos^4x)\ dx`
Now let `sin^2x=t implies 2sinxcosxdx=dt implies sinxcosxdx=(dt)/2`
`2I=pi/2 int_1^1 1/(t^2x+(1-t)^2) dx`
`I=pi/8 int_0^1 1/(2t^2-2t+1) dt`
...
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