1. **State the problem:** We need to find the indefinite integral of the function $3x + e^x$ with respect to $x$. 2. **Formula used:** The integral of a sum is the sum of the integrals: $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$$ Also, recall the basic integrals: - $$\int x \, dx = \frac{x^2}{2} + C$$ - $$\int e^x \, dx = e^x + C$$ 3. **Apply the integral:** $$\int (3x + e^x) \, dx = \int 3x \, dx + \int e^x \, dx$$ 4. **Integrate each term:** $$\int 3x \, dx = 3 \int x \, dx = 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}$$ $$\int e^x \, dx = e^x$$ 5. **Combine results and add constant of integration:** $$\int (3x + e^x) \, dx = \frac{3x^2}{2} + e^x + C$$ **Final answer:** $$\boxed{\frac{3x^2}{2} + e^x + C}$$