I happen to watch the video here,
which gives a solution to the definite integral below using the power series approach. Then answer is $\frac{\pi^2}{6}$, given by:
$$\int_0^1 \frac{\ln x}{x-1}dx=\int_{-1}^0 \frac{\ln(1+u)}{u}du=\sum_{n=0}^{\infty}\frac{1}{(n+1)^2}=\frac{\pi^2}{6},$$
where the power seires expansion of the function $\ln(1+u)$ is used.
I tried for some time, but could not find another approach. Does anyone know any alternative methods to evaluate above definite integral without using the infinite series expansion?
Any comments, or ideas, are really appreciated.
Best Answer
Here is to integrate without resorting to power series. Note \begin{align} \int_0^1\frac{\ln x}{1-x}dx& =\frac43\int_0^1 \frac{\ln x }{1-x}dx -\frac13\int_0^1 { \frac{\ln x }{1-x} } \overset{x\to x^2}{dx} \\ &= \frac43\int_0^1 \frac{\ln x}{1-x^2}dx = \frac23\int_0^\infty \frac{\ln x}{1-x^2}dx=\frac23J(1) \end{align}
where $ J(\alpha) =-\frac 12 \int_0^\infty \frac{\ln (1-\alpha^2 + \alpha^2 x^2)}{x^2-1}dx $
$$ J'(\alpha) =-\int_0^\infty \frac{\alpha dx}{1-\alpha^2 + \alpha^2 x^2} = -\frac{\pi/2}{\sqrt{1-\alpha^2}}$$
Thus
$$ \int_0^1\frac{\ln x}{1-x}dx = \frac23\int_0^1 J'(\alpha) d\alpha =-\frac{\pi}{3}\int_0^1 \frac{d\alpha}{\sqrt{1-\alpha^2}}= -\frac{\pi^2}{6}$$