1. **State the problem:** We want to understand why $\int \cos 5x \, dx = \frac{1}{5} \sin 5x + C$. 2. **Recall the integral rule:** The integral of $\cos(ax)$ with respect to $x$ is given by $$\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C,$$ where $a$ is a constant. 3. **Why this rule works:** When differentiating $\sin(ax)$, by the chain rule, $$\frac{d}{dx} \sin(ax) = a \cos(ax).$$ 4. **Adjusting for the constant $a$:** To get just $\cos(ax)$ when differentiating, we multiply $\sin(ax)$ by $\frac{1}{a}$: $$\frac{d}{dx} \left( \frac{1}{a} \sin(ax) \right) = \cos(ax).$$ 5. **Apply to our problem:** Here, $a = 5$, so $$\int \cos 5x \, dx = \frac{1}{5} \sin 5x + C.$$