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Thread: An inequality

  1. #1
    MHF Contributor

    May 2008

    An inequality

    The upper bound given in this problem is a good approximation for the integral:

    Problem: Prove that $\displaystyle \int_0^1 e^{\cos x} \ dx \leq e \sqrt{2} \tan^{-1}(\sqrt{2}/2).$
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  2. #2
    Senior Member Sampras's Avatar
    May 2009
    By the Rectangle Rule, $\displaystyle \int_{0}^{1} e^{\cos x} \ dx \approx e^{\cos \frac{1}{2}} $. Or we know that $\displaystyle e^{\cos x} $ has a bounded derivative over $\displaystyle [0,1] $. Now by the MVT, $\displaystyle xf'(y_x) = f(x)-f(0) $ for some $\displaystyle y_x \in [a,x] $. Also, $\displaystyle x<1 $. Then

    $\displaystyle \left|\int_{0}^{1} e^{\cos x} \ dx -e^{\cos 1} \right| = \left|\int_{0}^{1} x f'(y_x) \ dx \right| $

    $\displaystyle \left|\int_{0}^{1} e^{\cos x} \ dx -e^{\cos 1} \right| \leq \frac{1}{2} \sup_{0 \leq x \leq 1} |-e^{\cos x} \sin x| $
    Last edited by Sampras; Jul 6th 2009 at 11:06 AM.
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  3. #3
    Mar 2009
    OK, I have come up with an incredibly contrived solution suggested by the answer. Clearly it is equivalent to

    $\displaystyle \int_0^1\mathrm e^{\cos x-1}\mathrm dx\leq\int_0^1\frac 2{2+x^2}\mathrm dx$, so this suggests the inequality $\displaystyle \mathrm e^{\cos x-1}\leq\frac 2{2+x^2}$. If there's a better way...

    Here we go. Start with $\displaystyle \sin x\leq x$ and $\displaystyle \sin^2 x+\cos x=1+\cos x(1-\cos x)\geq 1$ on $\displaystyle [0,1]$. Therefore $\displaystyle \sin x\leq x(\sin^2 x+\cos x)$.

    This rearranges to $\displaystyle \frac{\sin x-x\cos x}{\sin^2 x}\leq x$. The LHS is the derivative of $\displaystyle \frac x{\sin x}$, so integrate both sides from $\displaystyle 0$ to $\displaystyle t$, where $\displaystyle t\in[0,1]$.

    Thus $\displaystyle \frac t{\sin t}-1\leq \frac{t^2}2$ and so $\displaystyle \sin t\geq\frac t{1+t^2/2}$. Integrate this from $\displaystyle 0$ to $\displaystyle x$ and $\displaystyle 1-\cos x\geq \ln(1+x^2/2)$ pops up.

    Guess what? $\displaystyle \cos x-1\leq\ln\left(\frac2{2+x^2}\right)$, and we arrive at our destination. Where's my hat? Rabbit, anyone?
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