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Math Help - Integral inequality

  1. #1
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    Integral inequality

    Let f be a function such that f(0)=0 and |f'(x)|\le\frac1{1+x} for x\ge0. Prove that \int_0^{e-1}\big(f(x)\big)^2\,dx\le e-2.
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  2. #2
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    \left| f(x) \right|=\left| \int_{0}^{x}{f'(t)\,dt} \right|\le \int_{0}^{x}{\left| f'(t) \right|\,dt}\le \int_{0}^{x}{\frac{dt}{1+t}}=\ln (1+x).

    Finally,

    \int_{0}^{e-1}{{{\left| f(x) \right|}^{2}}\,dx}\le \int_{0}^{e-1}{{{\ln }^{2}}(1+x)\,dx}=e-2.\quad\blacksquare
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