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Evaluate: int0^(pi//2)(xsinx cos\ x)/(si...

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

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`int_0^(pi//2)(xsinx cos\ x)/(sin^4x+cos^4x)dx`
`=int_0^(pi//2) frac{(pi/2 -x)sin(pi/2 -x)cos(pi/2 -x)}{sin^4(pi/2 -x)+cos^4(pi/2 -x)}dx`
`=int_0^(pi//2) frac{(pi/2 -x)cosx sinx}{cos^4x+sin^4x}`
`I=int_0^(pi//2)frac{cosx sinx}{sin^4x+cos^4x}dx-int_0^(pi//2) frac{xsinx. cosx}{sin^4x+cos^4x}dx`
`I=int_0^(pi//2)frac{cosx sinx}{sin^4x+cos^4x}dx-I`
`=pi/(2 xx 2) int_0^(pi/2) frac{2tanx.sec^2x}{1+(tan^2x)^2}dx`
`tan^x=z=>2tanx sec^2x dx=dz`
`x=0, z=0 and x=pi/2, z=oo` ...
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Evaluate the following integral: int_0^(pi//2)(xsinxcosx)/(sin^4x+cos^4x)dx

Prove that : int_(0)^(pi//2) (x sin x cos x)/(sin^(4) x+ cos^(4)x)dx =(pi)/(16)

Knowledge Check

  • int_(0)^(pi//2) (x sin x cos x)/(sin^(4) x + cos^(4)x) dx =

    A
    `(pi^(2))/(8)`
    B
    `(pi^(2))/(16)`
    C
    `(pi^(2))/(32)`
    D
    `(pi^(2))/(4)`

  • The value of int_(0)^(pi//2) (sin^(3)x cos x)/(sin^(4)x+ cos^(4)x )dx is

    A
    `pi//8`
    B
    `pi//4`
    C
    `pi//2`
    D
    `pi`

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    Evaluate: int_0^(pi/2) (sin2x)/(sin^4x+cos^4x)dx

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