Find the antiderivative: \( \int (3x^2 + 2x) \, dx = x^3 + x^2 + C \).
["How to Find the Antiderivative: A Complete Guide to ( \int (3x^2 + 2x) , dx = x^3 + x^2 + C )", "Calculating antiderivatives is a fundamental skill in calculus, essential for solving problems in physics, engineering, economics, and beyond. One of the most common and straightforward integrals you’ll encounter is finding the antiderivative of a polynomial, such as:", "[\n\int (3x^2 + 2x) , dx = x^3 + x^2 + C\n]", "This guide explains step-by-step how to arrive at this elegant result, why the process works, and why understanding antiderivatives matters.", "---", "### What Does Antiderivative Mean?", "Before diving in, it’s helpful to clarify:\nAn antiderivative of a function ( f(x) ) is another function ( F(x) ) such that\n[\nF'(x) = f(x)\n]\nFinding antiderivatives is essentially the reverse process of differentiation — a cornerstone of integration.", "---", "### Step-by-Step: Finding the Antiderivative", "We want to compute\n[\n\int (3x^2 + 2x) , dx\n]", "#### Step 1: Apply the Power Rule for Integration", "The power rule for integration states that for any linear term ( x^n ), where ( n <br/>\ne -1 ):", "[\n\int x^n , dx = \frac{x^{n+1}}{n+1} + C\n]", "We apply this term by term.", "#### Term 1: Integrate ( 3x^2 )", "Using the power rule:", "[\n\int 3x^2 , dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3\n]", "#### Term 2: Integrate ( 2x )", "Similarly,", "[\n\int 2x , dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2\n]", "---", "#### Step 2: Combine Results and Add the Constant of Integration", "Add the two results and include the constant ( C ), since antiderivatives are not unique and differ only by an additive constant:", "[\n\int (3x^2 + 2x) , dx = x^3 + x^2 + C\n]", "---", "### Why Is ( C ) Necessary?", "The constant ( C ) accounts for all vertical shifts in the family of functions that are derivatives of each other. For example, both ( x^3 + x^2 + 5 ) and ( x^3 + x^2 - 10 ) have the same derivative ( 3x^2 + 2x ). Therefore, ( C ) ensures the general solution is complete.", "---", "### Verifying the Result", "To confirm our answer, differentiate ( x^3 + x^2 + C ) and verify it matches the original integrand:", "[\n\frac{d}{dx}(x^3 + x^2 + C) = 3x^2 + 2x\n]", "This matches the integrand, confirming the solution is correct.", "---", "### Practical Applications", "Understanding antiderivatives like this supports many real-world applications:", "- Compute area under curves by definite integrals\n- Model physical quantities such as displacement from velocity\n- Solve optimization problems in economics and engineering\n- Analyze growth processes in biology and finance", "---", "### Summary", "Finding the antiderivative of ( 3x^2 + 2x ) is a foundational calculus task that reinforces key principles:", "- Apply the power rule term by term\n- Include the constant of integration\n- Verify your result by differentiation", "The final answer is:", "[\n\boxed{ \int (3x^2 + 2x) , dx = x^3 + x^2 + C }\n]", "With this method, you can confidently compute similar integrals — feel free to explore integrals of polynomials, trigonometric functions, and beyond!", "---", "Bonus Tip: The indefinite integral ( x^3 + x^2 + C ) is just one form; the result could also be written as\n[\nx^3 + x^2 + 5, \quad x^3 + x^2 + \ln|x| + C, \quad \ ext{etc.}\n]\nBut regardless of constants, demonstrated form matches the expected antiderivative.", "Start practicing — integration mastery opens doors in advanced math and science!"]
