I’m trying to understand core concepts about the closure of the set of elementary functions. Specifically, i would like to adress the following problem:
Find the all the numbers $k\in\mathbb{Z}$ such that the function $F$ defined by the indefinite integral: $$F(x)=\int \text{sin}(x)x^kdx$$ is elementary.
When $k=0$ the integral is just the sine integral. For the case $k=1$ is also a trivial matter, as: $$\int \text{sin}(x)xdx=\text{sin}(x)-\text{cos}(x)x+c$$ Thus, it’s clearly elementary. Repeated integration by parts lets you find similar solutions for $k\geq 0$ in the form: $$F(x)=p(x)\sin(x)+q(x)\cos(x)+c$$ where $\{p;q\}\in \mathbb{Z}[x]$. Thus, $F$ is elementary for $k\geq 0$. However, i cannot prove anything for $k<0$, as that exceeds my skills on calculus. I know from other sources that the case $k=-1$ is not elementary, and my intuition says is the same for every $k\leq 0$…but, how to prove such a thing?