\int_0^{45} (60 - (X + 15))\, dX = \int_0^{45} (45 - X)\, dX = \left[45X - \frac{X^2}{2}\right]_0^{45} = 45 \cdot 45 - \frac{45^2}{2} = 2025 - \frac{2025}{2} = \frac{2025}{2}

["Simplifying and Solving the Integral: A Step-by-Step Explanation", "When faced with an integral like\n[\n\int_0^{45} \left(60 - (X + 15)\right), dX,\n]\nit can feel daunting at first—especially when simplifying it step-by-step. But breaking the problem down reveals how powerful and elegant integration can be.", "---", "### Step 1: Simplify the Integrand", "Start by simplifying the expression inside the integral:\n[\n60 - (X + 15) = 60 - X - 15 = 45 - X.\n]\nSo the integral becomes:\n[\n\int_0^{45} (45 - X), dX.\n]\nThis clarity is essential—once we simplify, we're working with a straightforward linear function over a closed interval.", "---", "### Step 2: Apply the Antiderivative", "The antiderivative of (45 - X) is:\n[\n\int (45 - X), dX = 45X - \frac{X^2}{2} + C,\n]\nwhere (C) is the constant of integration. Since we are computing a definite integral, we can ignore the constant.", "---", "### Step 3: Evaluate at the Bounds", "Now plug in the upper limit (X = 45) and the lower limit (X = 0):\n[\n\left[45X - \frac{X^2}{2}\right]_0^{45} = \left(45 \cdot 45 - \frac{45^2}{2}\right) - \left(45 \cdot 0 - \frac{0^2}{2}\right).\n]\nSimplify:\n[\n= \left(2025 - \frac{2025}{2}\right) - 0 = \frac{4050 - 2025}{2} = \frac{2025}{2}.\n]", "---", "### Final Answer", "Thus,\n[\n\boxed{\int_0^{45} \left(60 - (X + 15)\right), dX = \int_0^{45} (45 - X), dX = \frac{2025}{2} = 1012.5}.\n]", "This integral, though simple in structure, demonstrates how algebraic simplification and fundamental calculus techniques work together to yield clear, accurate results. Whether in physics, engineering, or economics, mastering such integrals helps solve real-world problems involving area, cumulative growth, or average values over intervals.", "---", "Key Takeaways:\n- Simplify expressions before integration.\n- Use the power rule and linear antiderivatives effectively.\n- Evaluate at bounds carefully to avoid common mistakes.", "For further learning, explore definite integrals with real-world applications or practice with different linear and polynomial functions."]