Integral convergence with partial fraction

Question

I think $$ \int_2^\infty \frac{2x+3}{\left(x^2−1\right)x^2}\ dx $$ should converge,but after i use partial fraction, i get the following bunch of logarithms and the limit of those diverges: $$ \frac3x - \frac{\ln(|x+1|) + 4 \ln (|x|) - 5\ln(|x-1|)}{2} + C $$

Answer 1

The limit of the numerator with the logarithms is an indeterminate form $\infty-\infty$, so you cannot simply conclude that it diverges. You have to investigate further. Note that for $x\geq 2$, you can remove the absolute value signs. $$\begin{align} \ln(x+1)+4\ln(x)-5\ln(x-1)&=\ln(x+1)+\ln\left(x^4\right)-\ln\left(\left(x-1\right)^5\right)\\ &=\ln\left(\frac{(x+1)x^4}{(x-1)^5}\right) \\ &=\ln\left(\frac{x^5+x^4}{(x-1)^5}\right) \\ &=\ln\left(\frac{1+\frac{1}{x}}{\left(1-\frac{1}{x}\right)^5}\right) \end{align}$$ which converges to $\ln(1)=0$ as $x\to\infty$. And $\frac{3}{x}\to 0$ is clear. So the integral is actually convergent and is equal to $-F(2)$ where $F(x)$ is the antiderivative you obtained.