If f and g are defined as f(x) = f (a -x) and g(x) + g (a -x) = 4, then ∫0a f(x)g(x)dx is equal to

If f and g are defined as f(x) = f (a -x) and g(x) + g (a -x) = 4, then 0af(x)g(x)dx is equal to

  1. A

    0af(x)dx

  2. B

    20af(x)dx

  3. C

    20af(x)g(x)dx

  4. D

    None of these

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    Solution:

    Let,

    I=0af(x)g(x)dx----iI=0af(ax)g(ax)dx0af(x)dx=0af(ax)dx

    I=0af(x){4g(x)}dx---ii[f(x)=f(ax) and g(x)+g(ax)=4 (given) ]

    On adding Eqs. (i) and (ir), we get

    2I=0a4f(x)dxI=20af(x)dx

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