Consider the definite integral $$\int_{1/2}^x\cot(\pi x)~dx,$$ where $x>1$. As $x\rightarrow 1$ from below $\cot(\pi x)\rightarrow-\infty$, but as $x\rightarrow 1$ from above $\cot(\pi x)\rightarrow+\infty$. My question is this:
Could the definite integral in fact converge for $x>1$ so long as the rate at which the infinities either side of $x=1$ are reached are the same?
Furthermore, is it possible for $$\int_0^x\cot(\pi x)~dx$$ to converge, where $0<x<1$, even though $\lim\limits_{x\to0}= +\infty$ ?