Prove that $\displaystyle \int_0^1\ln\left(1+\sqrt x\right)\,dx=\int_0^1(1-x)\,dx$

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- Dec 13th 2007, 06:53 AMliyiIntegration proof
Prove that $\displaystyle \int_0^1\ln\left(1+\sqrt x\right)\,dx=\int_0^1(1-x)\,dx$

- Dec 13th 2007, 07:52 AMThePerfectHacker
- Dec 13th 2007, 08:54 AMKrizalid
I think liyi is not askin' to solve explicitly the integral. Here's a trick to prove the desired.

$\displaystyle \int_0^1 {\ln \left( {1 + \sqrt x } \right)\,dx} = \int_0^1 {\int_0^{\sqrt x } {\frac{1}

{{1 + u}}\,du} \,dx} = \int_0^1 {\underbrace {\int_{u^2 }^1 {dx} }_{\left( {1 - u^2 } \right)}\frac{1}

{{1 + u}}\,du} = \int_0^1 {(1 - u)\,du} \,\blacksquare$