1. **State the problem:** We need to find the integral of the function $5x$ with respect to $x$. 2. **Recall the formula:** The integral of $x^n$ with respect to $x$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration and $n \neq -1$. 3. **Apply the formula:** Here, $5x$ can be written as $5 \cdot x^1$. Using the constant multiple rule, $$\int 5x \, dx = 5 \int x^1 \, dx$$ 4. **Integrate:** Using the power rule, $$5 \int x^1 \, dx = 5 \cdot \frac{x^{1+1}}{1+1} + C = 5 \cdot \frac{x^2}{2} + C$$ 5. **Simplify:** $$= \frac{5}{2} x^2 + C$$ **Final answer:** $$\int 5x \, dx = \frac{5}{2} x^2 + C$$