Suppose I want to study the convergence of this integral: $$\int_{0}^{1}\frac{e^{\frac{i}{t(1-t)}}}{1+t}dt$$
My first approach was to develop the $e^{\frac{i}{t(1-t)}}$ into cos/sin form, thus giving me:
$$\int_{0}^{1}\frac{e^{\frac{i}{t(1-t)}}}{1+t}dt = \int_{0}^{1}\cos(\frac{1}{t(1-t)})dt + i\int_{0}^{1}\sin(\frac{1}{t(1-t)})dt$$
In both cases, $\cos(\frac{1}{t(1-t)})$ and $\sin(\frac{1}{t(1-t)})$ have no limit whenever $t \rightarrow 0$ or $t \rightarrow 1$.
Hint: $$ \left|\,\frac{e^{\frac i{t(1-t)}}}{1+t}\,\right| =\frac1{1+t} $$