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int0^(pi/2) (xsinxcosx)/(cos^4x+sin^4x) ...

`int_0^(pi/2) (xsinxcosx)/(cos^4x+sin^4x) dx=`

Text Solution

Verified by Experts

`"Let " I= int _(0)^(pi//4) (sinx cos x)/(cos^(4) x+sin^(4)x)dx`
`=int_(0)^(pi//4)(sin xcosx)/((1-sin^(2)x)^(2)+(sin^(2)x)^(2))dx`
`underset(" and " x =(pi)/(4) rArr t=(1)/(2))underset(" and "x=0 rArr t=0)("Let "" "sin^(2)x=t)`
`:. I=int_(0)^(1//2) (1)/((1-t)^(2)+t^(2)).(dt)/(2)`
`=(1)/(2) int_(0)^(pi//2) (1)/((1-t)^(2)+t^(2))dt`
`=(1)/(2) int_(0)^(1//2)(1)/(2t^(2)-2t+1)dt`
`=(1)/(2) .(1)/(2) int_(0)^(1//2) (1)/(t^(2)-t+(1)/(2))dt`
`=(1)/(4) int_(0)^(1//2) (1)/(t^(2)-t+((1)/(2))^(2)-((1)/(2))^(2)+(1)/(2))dt`
`=(1)/(4).(1)/(1)[tan^(-1)((t-(1)/(2))/((1)/(2)))]_(0)^(1//2)`
`=(1)/(2) [tan^(-1) 0-tan^(-1) (-1)]=(1)/(2).(pi)/(4)`
`=(pi)/(8)`
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