If ∫0π xf(sin⁡x)dx=A∫0π/2 f(sin⁡x)dx ,then A  is

If 0πxf(sinx)dx=A0π/2f(sinx)dx ,then A  is

  1. A

    2π

  2. B

    π

  3. C

    π/4

  4. D

    0

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    Solution:

    Let I=0πxf(sinx)dx

    I=0π(πx)f(sin(πx))dx         0af(x)dx=0af(ax)dxI=π0πf(sinx)dxII=π20πf(sinx)dx

    I=π2×20π/2f(sinx)dx                         02af(x)dx=20af(x)dx if, f(2ax)=f(x)I=π0π/2f(sinx)0πxf(sinx)dx=A0π/2f(sinx)dxπ0π/2f(sinx)dx=A0π/2f(sinx)dx=A=π

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