$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using the squeeze theorem using $-1\leq \sin(t)^m\leq 1$ and this resulted in $\frac{-1}{m}\leq\psi_m(x)\leq\frac{1}{m}$, which isn't useful.
Another part of the question was to determine the derivative. But I have no idea how, since the variable $x$, is in the limit of integration. I tried doing integration by parts, in hope of finding recursion, but this didn't really work (integrated by parts twice).
Any ideas how to solve this?