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
Here are the solutions to your questions. 1) Evaluate _-1^3 (3x^2 - 2x)\,dx Step 1: Find the indefinite integral of the function. (3x^2 - 2x)\,dx = 3(x^2+12+1) - 2(x^1+11+1) + C = 3((x^3)/(3)) - 2((x^2)/(2)) + C = x^3 - x^2 + C Step 2: Apply the limits of integration using the Fundamental Theorem of Calculus, F(b) - F(a). [x^3 - x^2]_-1^3 = ((3)^3 - (3)^2) - ((-1)^3 - (-1)^2) = (27 - 9) - (-1 - 1) = 18 - (-2) = 18 + 2 = 20 2) Express (9)/(1-4x) - (8)/(1-3x) as a single fraction. Step 1: Find a common denominator for the two fractions, which is (1-4x)(1-3x). Step 2: Rewrite each fraction with the common denominator. (9)/(1-4x) - (8)/(1-3x) = (9(1-3x))/((1-4x)(1-3x)) - (8(1-4x))/((1-4x)(1-3x)) Step 3: Combine the numerators over the common denominator. = (9(1-3x) - 8(1-4x))/((1-4x)(1-3x)) Step 4: Expand and simplify the numerator. = (9 - 27x - (8 - 32x))/((1-4x)(1-3x)) = (9 - 27x - 8 + 32x)/((1-4x)(1-3x)) = ((9 - 8) + (-27x + 32x))/((1-4x)(1-3x)) = (1 + 5x)/((1-4x)(1-3x)) Send me the next one 📸