Evaluate the integral \( \int (3x^2 - 4x + 1) \, dx \).
["# Evaluate the Integral ( \int (3x^2 - 4x + 1) , dx ): A Step-by-Step Guide", "When studying calculus, one of the foundational skills is learning how to evaluate integrals, especially polynomial functions. This article provides a comprehensive evaluation of the integral:", "[\n\int (3x^2 - 4x + 1) , dx\n]", "## Understanding the Integral", "The expression inside the integral is a quadratic polynomial:\n[\n3x^2 - 4x + 1\n]\nTo find its indefinite integral, we apply the power rule for integration, which states that:", "[\n\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad \ ext{(for } n <br/>\neq -1\ ext{)}\n]\nAdditionally, the integral of a sum is the sum of the integrals, so we can integrate each term separately.", "---", "## Step-by-Step Integration", "We start with:", "[\n\int (3x^2 - 4x + 1) , dx = \int 3x^2 , dx - \int 4x , dx + \int 1 , dx\n]", "### 1. Integrate ( 3x^2 )\nUsing the power rule:\n[\n\int 3x^2 , dx = 3 \int x^2 , dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3\n]", "### 2. Integrate ( -4x )\nFactor out the constant:\n[\n\int -4x , dx = -4 \int x , dx = -4 \cdot \frac{x^{1+1}}{1+1} = -4 \cdot \frac{x^2}{2} = -2x^2\n]", "### 3. Integrate ( 1 )\nIntegrating a constant:\n[\n\int 1 , dx = x\n]", "---", "## Combine All Terms", "Now, combine all the results:", "[\n\int (3x^2 - 4x + 1) , dx = x^3 - 2x^2 + x + C\n]", "where ( C ) is the constant of integration — essential for definite integrals or solutions to differential equations.", "---", "## Final Answer", "[\n\boxed{ \int (3x^2 - 4x + 1) , dx = x^3 - 2x^2 + x + C }\n]", "---", "## Why This Matters", "This integral demonstrates key concepts:\n- Applying the power rule to polynomial terms\n- Linearity of integration\n- Proper handling of constants\n- Forming the general antiderivative, including ( C )", "Mastering such integrals forms the basis for solving more complex problems in physics, engineering, and applied mathematics.", "---", "## SEO-Optimized Keywords & Phrases", "- "#integrate (3x^2 - 4x + 1)"\n- "#indefinite integral of (3x^2 - 4x + 1)"\n- "#antiderivative of (3x^2 - 4x + 1)"\n- "#solve ( \int (3x^2 - 4x + 1) , dx )"\n- "#calculus integration techniques"\n- "#poly polynomial integral guide"", "---", "By following this step-by-step guide, students and learners can confidently evaluate and understand integrals of quadratic polynomials. For further practice, try integrating other polynomials or apply differentiation to verify your result."]
