TITLE: Integrals $ \int_0^1 \log x \mathrm dx $,$\int_2^\infty \frac{\log x}{x} \mathrm dx $,$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$
QUESTION [8 upvotes]: 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.

REPLY [8 votes]: Here's an alternative derivation for a), where improperness plays no role.
$$
\int_0^1 { - \log x\,{\rm d}x}  = \int_{x = 0}^1 {\int_{u = x}^1 {\frac{1}{u}\,{\rm d}u} \,{\rm d}x}  = \int_{u = 0}^1 {\int_{x = 0}^u {\frac{1}{u}\,{\rm d}x} \,{\rm d}u}  = \int_{u = 0}^1 {\frac{1}{u}u\,{\rm d}u}  = 1.
$$