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.