I am trying to prove that $\lim_{x\to0} \int_{x}^{1 \over x} {(\cos 2r-\cos r) \over r} dr$ exists. Wolframalpha gives me the value ${\int_{0}^{\infty} {(\cos 2x-\cos x)\over x} dx} = -\ln2$, but I don't know where to start, based only on this information. The question is in the book "Real Analysis and Foundations" by S. Krantz.
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do you only want to prove the existence of the limit or also the value $-\log(2)$? – tired Nov 11 '16 at 11:35
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Does $\cos 2r - \cos r = -2\sin{3r\over2}\sin{r\over 2}$ help ? – Astyx Nov 11 '16 at 11:36
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2This is an example of a Frullani integral. See http://math.stackexchange.com/questions/61828/proof-of-frullanis-theorem – David H Nov 11 '16 at 11:41
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Have a look at this wonderful thread: http://math.stackexchange.com/questions/1807410/frullani-s-theorem-in-a-complex-context – Jack D'Aurizio Nov 11 '16 at 12:54
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$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\lim_{x \to 0^{+}}\ \int_{x}^{1/x}{\cos\pars{2r} - \cos\pars{r} \over r} \,\dd r:\ {\large ?}}$.
\begin{align} &\lim_{x \to 0^{+}}\ \int_{x}^{1/x}{\cos\pars{2r} - \cos\pars{r} \over r}\,\dd r = \int_{0}^{\infty}\bracks{\cos\pars{2r} - \cos\pars{r}}\ \overbrace{% \int_{0}^{\infty}\expo{-rt}\,\dd t}^{\ds{1 \over r}}\,\dd r \\[5mm] = &\ \Re\int_{0}^{\infty}\int_{0}^{\infty} \bracks{\expo{\pars{-t + 2\ic}r} - \expo{\pars{-t + \ic}r}}\dd r\,\dd t = \Re\int_{0}^{\infty}\pars{-\,{1 \over -t + 2\ic} + {1 \over -t + \ic}}\,\dd t \\[5mm] = &\ \left. \Re\ln\pars{t - 2\ic \over t - \ic} \right\vert_{\ t\ =\ 0}^{\ t\ \to\ \infty} =\ \bbox[10px,#ffe,border:1px dotted navy]{\ds{-\ln\pars{2}}} \end{align}
Felix Marin
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Hint:
$\lim_{x\to0} \int_{x}^{1 \over x} {(\cos 2r-\cos r) \over r} dr$
$=\lim_{x\to0} \int_{x}^{1 \over x} (\cos 2r-\cos r) d(\ln r)$
$=\lim_{x\to0} \int_{x}^{1 \over x} \cos 2r\, d(\ln r) - \int_{x}^{1 \over x} \cos r\,d(\ln r)$
Using integration by parts should help