The Riemann-Lebesgue Lemma via Integration by Parts

A rapidly oscillating factor like $\sin(nx)$ or $\cos(nx)$ averages any fixed integrable function to zero, so $\int_a^b f(x)\sin(nx)\,dx \to 0$ as $n\to\infty$. When $f'$ is continuous, integration by parts proves it and gives the $\Theta(1/n)$ decay rate.