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Thread: Definite Integration

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

    $\displaystyle F(x)$ is a differentiable function such that

    $\displaystyle F'(a-x)=F'(x)$

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

    Evaluate

    $\displaystyle \int_{0}^aF(x)dx$

    and give an example of such a function.
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  2. #2
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    Quote Originally Posted by pankaj View Post
    $\displaystyle F(x)$ is a differentiable function such that

    $\displaystyle F'(a-x)=F'(x)$

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

    Evaluate

    $\displaystyle \int_{0}^aF(x)dx$

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

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