How to show the below is an improper integral

Question

How would you prove the following integral diverges: $\int_0^1\theta^{-1}(1-\theta)^{c-1} d\theta $ for any constant c>0

Answer 1

$\begin{array}\\ \int_0^1t^{-1}(1-t)^{c-1} dt &\gt \int_0^{\frac12}t^{-1}(1-t)^{c-1} dt\\ &= \int_0^{\frac12}t^{-1}\frac{(1-t)^{c}}{1-t} dt\\ &\gt \int_0^{\frac12}t^{-1}(\frac12)^{c} dt\\ &= \frac1{2^c}\int_0^{\frac12}t^{-1} dt\\ \end{array} $

and this diverges because $\lim_{x \to 0}\int_x^{\frac12}t^{-1} dt =\lim_{x \to 0}(\ln(\frac12)-\ln(x)) \to \infty$.