Evaluate the integral \( \int (2x + 3) \, dx \).

["# Evaluate the Integral ( \int (2x + 3) , dx ): A Complete Guide", "When learning calculus, one of the most fundamental operations is integration. Among the many essential functions to integrate, linear expressions like ( \int (2x + 3) , dx ) appear frequently in both academic and applied contexts. Whether you're a student mastering basic integration techniques or a professional needing a quick review, this guide will walk you through evaluating the integral step-by-step, explain the reasoning behind the method, and highlight its applications.", "## Why Integrate ( 2x + 3 )?", "The expression ( \int (2x + 3) , dx ) represents the indefinite integral of a linear polynomial. Integrals are used in physics, engineering, economics, and computer science to model accumulated quantities—like total distance traveled given a velocity function or the area under a curve. Understanding how to compute such integrals improves problem-solving skills and builds a foundation for more advanced calculus topics, including differentiation under the integral sign, substitution methods, and applications in real-world modeling.", "## Step-by-Step Evaluation", "### Step 1: Recall the Power Rule for Integration", "The basic rule for integrating polynomial terms states:", "[\n\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad \ ext{(for } n <br/>\ne -1\ ext{)}\n]", "This rule comes directly from reversing the power rule for differentiation.", "For linear expressions, we apply linearity of the integral:", "[\n\int (a + bx) , dx = \int a , dx + \int bx , dx\n]", "Here, ( a = 3 ) and ( b = 2 ).", "### Step 2: Apply Linearity", "Separate the integral into constant and linear parts:", "[\n\int (2x + 3) , dx = \int 2x , dx + \int 3 , dx\n]", "### Step 3: Integrate Each Term", "- For ( \int 2x , dx ):\n Factor out the constant ( 2 ):\n [\n 2 \int x , dx = 2 \cdot \frac{x^{2}}{2} = x^2\n ]", "- For ( \int 3 , dx ):\n Treat 3 as ( 3x^0 ):\n [\n \int 3 , dx = 3x + C_1\n ]", "### Step 4: Combine Results", "Add the results and include the constant of integration ( C ):", "[\n\int (2x + 3) , dx = x^2 + 3x + C\n]", "This is the final evaluated integral.", "## Final Answer", "[\n\boxed{x^2 + 3x + C}\n]", "## Verification via Differentiation", "To confirm correctness, differentiate ( x^2 + 3x + C ):", "[\n\frac{d}{dx}(x^2 + 3x + C) = 2x + 3\n]", "Since this matches the original integrand, the integration is verified.", "## Applications of ( \int (2x + 3) , dx )", "- Area under a Straight Line: Represents accumulated value of a constant slope with an initial offset.\n- Motion Analysis: If ( 2x ) is velocity, the integral gives total displacement with a linear velocity term.\n- Cost and Revenue Models: Linear cost and linear revenue terms combined help compute net change over variable quantities.", "## Alternative Methods: Substitution (Optional Insight)", "Although not necessary here, substitution illustrates integration mastery. Recognize that ( 2x + 3 ) transforms easily under substitution, but direct integration via linearity remains most efficient for linear functions.", "## Conclusion", "Evaluating ( \int (2x + 3) , dx ) exemplifies core integration skills: applying linearity, leveraging power rules, and working with constant multiples. Mastering such basic integrals equips learners for advanced integration techniques and real-world calculations. Whether you're solving a problem in physics, economics, or engineering, knowing how to integrate linear expressions is both practical and profoundly useful.", "Remember: Always double-check by differentiating your result to ensure accuracy.", "---", "Keywords: integrate ( 2x + 3 ), indefinite integral, ( \int (2x + 3) dx ), linear function integration, calculus tutorial, integration method, verify derivative, foundational calculus."]