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Evaluate int(0)^(pi//2)(xsinxcosx)/(cos^...

Evaluate `int_(0)^(pi//2)(xsinxcosx)/(cos^(4)x+sin^(4)x)dx`

Text Solution

Verified by Experts

The correct Answer is:
`(pi^(2))/(16)`
Let `I= int_(0)^(pi//2) (x sin x* cos x)/(cos^(4)x + sin ^(4) x)dx`
`rArr I = int_(0)^(pi//2) (((pi)/(2)-x)sin ((pi)/(2)-x)* cos ((pi)/(2)-x))/(sin ^(4)((pi)/(2)-x)cos ^(4) ((pi)/(2)-x))dx`
`rArr I= int _(0)^(pi//2) (((pi)/(2)x)* sin xcos x)/(cos^(4)x+ sin ^(4)x)dx`
`rArr I = (pi)/(2) int_(0)^(pi//2)(sin x cos x)/(sin^(4) x + cos^(4) x)dx- int_(0)^(pi//2)(x sin x * cos x)/( sin^(4) x + cos^(4) x)dx`
`= (pi)/(2) int_(0)^(pi//2) ( sin x * cosx)/( sin^(4) x + cos^(4)) dx-I`
` rArr 2 I = (pi)/(2) int _(0)^(pi//2) (tan x * sin ^(2)x)/(tan^(4) x + 1)dx`
` rArr 2 I = (pi)/(2)* (1)/(2) int_(0)^(pi//2) (1)/((tan ^(2) x)^(2))d (tan^(2)x)`
` rArr 2 I = (pi)/(4)* [ tan ^(-1) t]_(0)^(oo) = (pi)/(4)(tan ^(-1)oo - tan ^(-1) 0)` [ where , t ` tan^(2) x]`
` rArr I= (pi^(2))/(16)`
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