Compute the integral \(\int (3x^2 - 2x + 1) \, dx\).
["Compute the Integral (\int (3x^2 - 2x + 1) , dx): A Step-by-Step Guide", "Integration is a fundamental concept in calculus, playing a crucial role in mathematics, physics, engineering, and many other fields. One of the most common integrals students encounter is computing the indefinite integral of polynomial functions. In this article, we will walk through the process of computing the integral (\int (3x^2 - 2x + 1) , dx) in detail, explaining each step clearly and highlighting key techniques involved.", "---", "### Understanding the Integral", "We are tasked with finding:", "[\n\int (3x^2 - 2x + 1) , dx\n]", "This is the indefinite integral of the quadratic function (3x^2 - 2x + 1). The result will be a family of functions annotated with the constant of integration (C), since indefinite integrals always include a constant.", "---", "### Step 1: Apply the Power Rule for Integration", "The power rule for integration states that for any real number (n <br/>\neq -1),", "[\n\int x^n , dx = \frac{x^{n+1}}{n+1} + C\n]", "We apply this rule term-by-term to the polynomial inside the integral.", "---", "### Step 2: Integrate Each Term", "Break the integral into three separate integrals:", "[\n\int (3x^2 - 2x + 1) , dx = 3\int x^2 , dx - 2\int x , dx + \int 1 , dx\n]", "Now integrate each term individually:", "1. (\int x^2 , dx = \frac{x^{3}}{3}) → by the power rule with (n = 2), so\n (3 \int x^2 , dx = 3 \cdot \frac{x^3}{3} = x^3)", "2. (\int x , dx = \frac{x^{2}}{2}) → thus\n (-2 \int x , dx = -2 \cdot \frac{x^2}{2} = -x^2)", "3. (\int 1 , dx = x) (since the integral of a constant (1) is (x + C))", "---", "### Step 3: Combine the Results", "Add the results of the individual integrations and include the constant (C):", "[\n\int (3x^2 - 2x + 1) , dx = x^3 - x^2 + x + C\n]", "This is the final antiderivative.", "---", "### Why This Method Works", "By using the linearity of integration and the foundational power rule, we efficiently break down a complex-looking polynomial into manageable pieces. This method ensures accuracy and clarity, making it a go-to strategy for integrating polynomials.", "---", "### Final Answer", "[\n\boxed{\int (3x^2 - 2x + 1) , dx = x^3 - x^2 + x + C}\n]", "---", "### Applications of the Integral", "While this integral is algebraically simple, the process illustrates broader applications:", "- Area Under a Curve: The definite integral from (a) to (b) gives the area under (y = 3x^2 - 2x + 1).\n- Physics: Used to calculate displacement from velocity functions.\n- Economics: Models cumulative growth or cost over time from rate functions.", "Understanding how to compute integrals like this strengthens skills for tackling real-world problems in science and engineering.", "---", "### Summary", "Computing (\int (3x^2 - 2x + 1) , dx) involves:", "- Applying the power rule to each term,\n- Multiplying coefficients correctly,\n- Adding the integration constant,\n- Writing the final result as a polynomial plus (C).", "With practice, this step-by-step approach becomes intuitive, enabling efficient integration of more complex expressions.", "If you're learning calculus, mastering such integrals is essential—keep practicing to build confidence and precision!"]
