This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Apply the power rule for integration to each term. The power rule for integration states that x^n\,dx = x^n+1n+1 + C. We apply this rule to each term in the polynomial. (2x^3-5x^2+4x-1)\,dx = 2x^3\,dx - 5x^2\,dx + 4x\,dx - 1\,dx Integrate each term: 2x^3\,dx = 2 · x^3+13+1 = 2 · (x^4)/(4) 5x^2\,dx = 5 · x^2+12+1 = 5 · (x^3)/(3) 4x\,dx = 4 · x^1+11+1 = 4 · (x^2)/(2) 1\,dx = x Step 2: Combine the integrated terms and simplify. (2x^3-5x^2+4x-1)\,dx = 2 · (x^4)/(4) - 5 · (x^3)/(3) + 4 · (x^2)/(2) - x + C = (2x^4)/(4) - (5x^3)/(3) + (4x^2)/(2) - x + C = (1)/(2)x^4 - (5)/(3)x^3 + 2x^2 - x + C The final answer is (1)/(2)x^4 - (5)/(3)x^3 + 2x^2 - x + C. 3 done, 2 left today. You're making progress.