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Math Help - Integral. #5

  1. #1
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    Integral. #5

    Problem :

    if f is continuous on [0,a] and f(x) \cdot f(a-x) = 1 , Evaluate :

    \int_0^a \dfrac{dx}{1+f(x)}.

    Final answer should be \dfrac{a}{2}
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  2. #2
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    Let I = \int_0^a \dfrac{dx}{1+f(x)}

    Make the change of variable x = a - u so your integral becomes

    I = - \int_a^0 \dfrac{du}{1+f(a-u)}

    Now your your f(x) \cdot f(a-x) = 1 relation

    so

    I = \int_0^a \dfrac{du}{1 + 1/f(u)} = \int_0^a \dfrac{f(u)}{1 + f(u)}du

    Now replace the u with x and add the two integrals

    2I =  \int_0^a \dfrac{dx}{1+f(x)} +  \int_0^a \dfrac{f(x)dx}{1+f(x)} = \int_0^a dx

    and solve for I.
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