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Math Help - Definite Integration

  1. #1
    Senior Member pankaj's Avatar
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    Definite Integration

    F(x) is a differentiable function such that

    F'(a-x)=F'(x)

    for all x satisfying 0\leq x\leq a.

    Evaluate

    \int_{0}^aF(x)dx

    and give an example of such a function.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by pankaj View Post
    F(x) is a differentiable function such that

    F'(a-x)=F'(x)

    for all x satisfying 0\leq x\leq a.

    Evaluate

    \int_{0}^aF(x)dx

    and give an example of such a function.
    f'(a-x)=f'(x) gives us f(x)+f(a-x)=c=\text{constant}. put x=a/2 to get c=2f(a/2). now substitute x \to a-x to get \int_0^af(x) \ dx = \int_0^a f(a-x) \ dx = \int_0^a (c-f(x)) \ dx.

    thus: \int_0^a f(x) \ dx = \frac{ac}{2}=af(a/2). an example is f(x)=x. here we have c=a and \int_0^a x \ dx = \frac{a^2}{2}=a \cdot a/2=af(a/2).
    Last edited by NonCommAlg; June 11th 2009 at 08:05 PM. Reason: unimportant typo!
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