A problem told by Zhivko Petrov
The following integral was proposed for homework to Applied Math, by Zhivko Petrov. $\textbf{Problem}.$ Evaluate $$\int_{0}^1\frac{\ln(1-x+x^2)}{x^2-x}\text{d} x.$$ Later I communicated the problem to prof. Gadjev and he proposed a neat solution (to be presented later). Now we elaborate a solution, on an idea proposed by prof. Babev. I would like to thank the afformentioned people, as well as David Petrov for pointing me to that idea. $\textbf{Solution}.$ Introduce $$I(y)=\int_0^1\frac{\ln(1-y(x-x^2))}{x^2-x}\text{d}x.$$ We need to find $I(1)$. Clearly $I(0)=0$. Using differentiation under the integral sign we obtain $$I'(y)=\int_0^1\frac{1}{1-y(x-x^2)}\text{d}x.$$ Assuming that $y\in [0,1]$, one can easily integrate the last to obtain $$I'(y)=\frac{4 \arcsin\left(\frac{\sqrt{y}}{2}\right)}{\sqrt{y(4-y) }}.$$ The latter is very easy to integrate (for example making the change $y=4t^2$) in order to obtain $$I'(y)=\left(4 \arcsin\left(\frac{\sqrt{y}}{2}\right)^2\right)