I'm trying a physics problem for fun and I ran into this integral:
$$ \int dv = \int \ln{\left(\frac{x}{x_0}\right)} \frac{1}{x-x_0} dx $$
Unfortunately I can almost do a u substitution, but not quite. So I did integration by parts:
$$ = \ln{\left(\frac{x}{x_0}\right)} \ln{(x-x_0)} - \int \ln{(x-x_0)} \frac1x dx $$
Unfortunately again the integral isn't solvable so I tried integration by parts yet again...
$$ = \ln{\left(\frac{x}{x_0}\right)} \ln{(x-x_0)} - \left[ \ln (x -x_0) \ln x - \int \ln x \frac{1}{x-x_0} dx \right]$$
Yet another integral I don't know how to solve so I tried one more futile attempt at integration by parts...
$$ = \ln{\left(\frac{x}{x_0}\right)} \ln{(x-x_0)} - \ln (x -x_0) \ln x + \left[ \ln x \ln (x-x_0) - \int \ln (x-x_0)\frac1x dx \right]$$
I suppose it wasn't completely useless to try and integrate it one last time because I recognized a pattern- that I'm seeing repeats of integrals and each time I integrate, it seems to produce new terms that cancels out previous terms. I plotted the numerical integral and it has a logarithmic shape.
Is there any solution to this integral? If so, can you walk me through it? Keep in mind I'm in Calc 2 as a reference for what I will be able to comprehend.