Integrate ∫ 6x dx.

**The reasoning** Integration is the reverse of differentiation. When we integrate 6x with respect to x, we use the **power rule for integration**: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C Here, we have 6x = 6x¹ So: ∫ 6x dx = 6 × (x¹⁺¹)/(1+1) + C = 6 × (x²/2) + C = **3x² + C** The **+ C is crucial** — it represents the constant of integration. Since the derivative of any constant is zero, when we reverse the process (integrate), we must account for all possible constants that could have been there originally. **Why the wrong options tempt you** - **Option A (3x²)**: You did the math correctly but forgot the +C. This is the most common mistake — never omit the constant in indefinite integrals! - **Option C (6x² + C)**: You forgot to divide by the new power (2). You kept the 6 but didn't apply the full power rule. - **Option D (6)**: You might have confused integration with differentiation. This would be if you differentiated 6x instead. **Quick takeaway** For indefinite integrals, **always add + C** — it's not optional, it's part of the complete answer!