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Math Help - Integration proof

  1. #1
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    Integration proof

    Prove that \int_0^1\ln\left(1+\sqrt x\right)\,dx=\int_0^1(1-x)\,dx
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  2. #2
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    Quote Originally Posted by liyi View Post
    Prove that \int_0^1\ln\left(1+\sqrt x\right)\,dx=\int_0^1(1-x)\,dx
     t = 1+\sqrt{x}
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  3. #3
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    I think liyi is not askin' to solve explicitly the integral. Here's a trick to prove the desired.

    Quote Originally Posted by liyi View Post
    Prove that \int_0^1\ln\left(1+\sqrt x\right)\,dx=\int_0^1(1-x)\,dx
    \int_0^1 {\ln \left( {1 + \sqrt x } \right)\,dx} = \int_0^1 {\int_0^{\sqrt x } {\frac{1}<br />
{{1 + u}}\,du} \,dx} = \int_0^1 {\underbrace {\int_{u^2 }^1 {dx} }_{\left( {1 - u^2 } \right)}\frac{1}<br />
{{1 + u}}\,du} = \int_0^1 {(1 - u)\,du} \,\blacksquare
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