Convergence of integral with parameter For which $t$ does the following integral converge?

$$\int_{0}^{1/2}\frac{(1-\cos x)^t(1-x)^3}{\ln^2(1-x)}\,dx$$ where $t\in \mathbb{R}$.
Hint. As $x$ tends to $0^+$, you have
$$\frac{(1-\cos x)^t(1-x)^3}{\ln^2(1-x)} \sim \frac{(x^2/2)^t}{x^2}=\frac1{2^t}x^{2t-2}$$ your integral converges iff $t&gt;\frac12$.
Hint: Use series. It is necessary to see first two terms in $(1-x)^3/\ln^2(1-x)$.