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...