Show that the improper integral of (x^(-p))(cos x)dx from 0 to 1 converges iff p < 1.
I assume that $\displaystyle 0 < p$ in order to make it improper.
Note that for $\displaystyle 0 \leq x \leq 1$ we have $\displaystyle 0\leq \cos x \leq 1 - \frac{x^2}{2!}+\frac{x^4}{4!}$.
This means, $\displaystyle 0\leq x^{-p} \cos x \leq x^{-p} \implies 0 \leq \int_0^1 x^{-p} \cos x ~ dx \leq \int_0^1 x^{-p} ~ dx = \frac{1}{1-p}$.
Why does it feel as it I did something wrong?