How should I study the convergence of this integral?
$$\int_{0}^{1}\frac{\cos(\frac{1}{t^{17}(1-t)^{13}})}{1+t}$$
I can state a few things:
$$\cos(\frac{1}{t^{17}(1-t)^{13}}) \sim \cos(\frac{1}{t^{17}})$$ when $x \rightarrow 0$
Thus it doesn't have a limit when $x \rightarrow 0$. So can I already state that the integral doesn't converge?
If $f(t)$ is the integrand, then $\left|f(t)\right|\le \dfrac{1}{t+1}$ for all $x\in (0,1),$ so the integral is absolutely convergent.