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I don't get how we're supposed to use analysis to calculate things like:

a) $$ \int_0^1 \log x \mathrm dx $$ b) $$\int_2^\infty \frac{\log x}{x} \mathrm dx $$ c) $$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$$

I tried integration by parts, but not helpful since we have it unbounded. Please explain how to do this using improper integrals in analysis.

  • 1
    For a), you could try the substitution $x=\exp(-u)$... b) looks divergent, and c) succumbs to the substitution $x=\tan(u)$.2011-04-28
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    For a) Integrate by parts $\int_{0}^{1}\log x\;\mathrm{d}x=\int_{0}^{1}1% \cdot \log x\;\mathrm{d}x$. Apply l'Hôpital to $\lim_{x\rightarrow 0^{+}}x\log x$.2011-04-28
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    b) is divergent for sure, either by comparison to integrating $1/x$, or by by contemplating the derivative of $(\log x)^2$.2011-04-28
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    One question per post please!2015-01-12

3 Answers 3