contour integral with integration by parts

Question

Is there a complex version of integration-by-part? I saw someone used it but didn't find it in textbook. I tested integrals $\int_{\mathcal{C}}\frac{\log(x+1)}{x-2}\mathrm{d}x$ and $\int_{\mathcal{C}}\frac{\log(x-2)}{x+1}\mathrm{d}x$, where $\mathcal{C}$ encloses both -1 and 2. But the results do not match. Is it because they are not equal at the first place or I chose the wrong branch cut?

Answer 1

Integration by parts is just the product rule and the Fundamental Theorem of Calculus. But you need well-defined analytic functions on your contour, which you don't have here.