1. We are asked to find the integral $\int \sin 5x \, dx$. 2. The formula for integrating sine functions with a linear argument is: $$\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$$ where $a$ is a constant and $C$ is the constant of integration. 3. Applying this formula with $a=5$: $$\int \sin 5x \, dx = -\frac{1}{5} \cos 5x + C$$ 4. This means the antiderivative of $\sin 5x$ is $-\frac{1}{5} \cos 5x + C$. Final answer: $$\boxed{-\frac{1}{5} \cos 5x + C}$$