Let \[ \varphi(p)\triangleq \int_0^\pi x^p \sin x\; dx \quad\text{and}\quad f(x,p)=x^p\sin x. \]Note that $\varphi(0)=2$ and $\varphi(1)=3$ (using integration by parts). Moreover, the integrand $f(x,p)$ is continuous over compact domain $[0,\pi]\times [0,1]$ so is uniformly continuous. From here, it follows $\varphi$ is continuous. Hence, by intermediate value theorem, for some $0<p<1$ it holds that $\varphi(p)=\sqrt[10]{2000}$ as $2^{10}<2000<3^{10}$.