Evaluate the definite integral \( \int_{0}^{2} (3x^2 + 2x) \, dx \).

["# Evaluate the Definite Integral ( \int_{0}^{2} (3x^2 + 2x) , dx )", "When studying calculus, evaluating definite integrals is a fundamental skill. One common example is computing integrals of polynomial functions over specific intervals. In this article, we’ll carefully evaluate the definite integral:", "[\n\int_{0}^{2} (3x^2 + 2x) , dx\n]", "## Why Evaluate Definite Integrals?", "Definite integrals represent the signed area under a curve between two bounds. Evaluating integrals helps solve real-world problems in physics, engineering, economics, and more. Understanding how to compute ( \int_{a}^{b} f(x),dx ) equips learners with tools to analyze motion, compute probabilities, calculate work, and beyond.", "## Step-by-Step Evaluation", "### Step 1: Find the Antiderivative", "The integrand is ( 3x^2 + 2x ). We begin by finding its indefinite integral (antiderivative):", "[\n\int (3x^2 + 2x) , dx = \int 3x^2 , dx + \int 2x , dx\n]", "Using basic integration rules:", "- ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ) for ( n <br/>\neq -1 )\n- Constant multiples can be factored out", "Compute each term:", "[\n\int 3x^2 , dx = 3 \cdot \frac{x^3}{3} = x^3\n]\n[\n\int 2x , dx = 2 \cdot \frac{x^2}{2} = x^2\n]", "So the antiderivative is:", "[\nF(x) = x^3 + x^2 + C\n]", "Since the constant ( C ) cancels out in definite integrals, we can ignore it.", "### Step 2: Apply the Fundamental Theorem of Calculus", "Now evaluate the antiderivative at the upper and lower limits:", "[\n\left[ x^3 + x^2 \right]<em 0="0">{0}^{2} = \left( (2)^3 + (2)^2 \right) - \left( (0)^3 + (0)^2 \right)\n]", "Calculate each term:", "- ( 2^3 = 8 )\n- ( 2^2 = 4 )\n- ( 0^3 = 0 ), ( 0^2 = 0 )", "[\n= (8 + 4) - (0 + 0) = 12\n]", "### Final Answer", "[\n\int (3x^2 + 2x) , dx = 12}^{2\n]", "This result means the net area under the curve ( y = 3x^2 + 2x ) from ( x = 0 ) to ( x = 2 ) is 12 square units.", "## Alternative Explanation: Area Under the Curve", "Graphically, this integral computes the signed area between the x-axis and the curve ( f(x) = 3x^2 + 2x ) over ( [0, 2] ). Since ( 3x^2 + 2x \geq 0 ) on this interval, all areas are positive, and the total is simply the sum of areas approximated by rectangles—without overcounting due to signs.", "## Summary", "Evaluating the definite integral ( \int_{0}^{2} (3x^2 + 2x) , dx ) involves:", "1. Finding the antiderivative ( x^3 + x^2 )\n2. Applying the limits to compute the net area\n3. Interpreting the result as the signed area under the curve", "(\boxed{12})", "---", "Keywords: evaluate definite integral, ( \int_{0}^{2} (3x^2 + 2x),dx ), antiderivative, fundamental theorem of calculus, area under curve, integral evaluation."]