Show that $\displaystyle \int_0^1\left(\ln\left(1+\frac{1}{x}\right)-\frac{1}{1+x}\right)\,dx$ converges.

Question

Show that $\displaystyle \int_0^1\left(\ln\left(1+\frac{1}{x}\right)-\frac{1}{1+x}\right)\,dx$ converges.

I think that the expression on the right is pretty trivial, but how do I show the "$\ln$" expression converges? both hints/solutions are welcome.

Answer 1

Your integral is$$\begin{align}\int_0^1\left(\ln(1+x)-\ln x-\frac{1}{1+x}\right)dx&=[x\ln(1+x)-x\ln x]_0^1\\&=\ln 2+\lim_{x\to0^+}x\ln x\\&=\ln 2.\end{align}$$