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
Another one m — let's solve it. Here are the solutions for the definite integrals using the substitution method. I) For the integral _0^6 (6x)/(3x-2)\,dx: Step 1: Define the substitution and find the differential. Let u = 3x-2. Then du = 3\,dx, which means dx = (1)/(3)\,du. Also, from u = 3x-2, we have 3x = u+2, so 6x = 2(u+2). Step 2: Change the limits of integration. When x=0, u = 3(0)-2 = -2. When x=6, u = 3(6)-2 = 18-2 = 16. Step 3: Substitute into the integral and simplify. _-2^16 (2(u+2))/(u) · (1)/(3)\,du = (2)/(3) _-2^16 (u+2)/(u)\,du = (2)/(3) _-2^16 (1 + (2)/(u))\,du Step 4: Integrate and evaluate using the new limits. (2)/(3) [u + 2|u|]_-2^16 (2)/(3) [(16 + 2|16|) - (-2 + 2|-2|)] (2)/(3) [16 + 2(16) + 2 - 2(2)] (2)/(3) [18 + 2((16) - (2))] (2)/(3) [18 + 2((16)/(2))] (2)/(3) [18 + 2(8)] (2)/(3) · 2 [9 + (8)] (4)/(3) [9 + (8)] = 12 + (4)/(3)(8) Since (8) = (2^3) = 3(2), the answer can also be written as: 12 + (4)/(3)(3(2)) = 12 + 4(2) The final answer is 12 + 4(2). II) For the integral _5^6 (1-2x)/(x-4)\,dx: Step 1: Define the substitution and find the differential. Let u = x-4. Then du = dx. Also, from u = x-4, we have x = u+4. So 1-2x = 1-2(u+4) = 1-2u-8 = -2u-7. Step 2: Change the limits of integration. When x=5, u = 5-4 = 1. When x=6, u = 6-4 = 2. Step 3: Substitute into the integral and simplify. _1^2 (-2u-7)/(u)\,du = _1^2 (-2 - (7)/(u))\,du Step 4: Integrate and evaluate using the new limits. [-2u - 7|u|]_1^2 [(-2(2) - 7|2|) - (-2(1) - 7|1|)] [-4 - 7(2)] - [-2 - 7(0)] -4 - 7(2) + 2 -2 - 7(2) The final answer is -2 - 7(2). III) For the integral _0^3 (2x-3)/(2x+8)\,dx: Step 1: Define the substitution and find the differential. Let u = 2x+8. Then du = 2\,dx, which means dx = (1)/(2)\,du. Also, from u = 2x+8, we have 2x = u-8. So 2x-3 = (u-8)-3 = u-11. Step 2: Change the limits of integration. When x=0, u = 2(0)+8 = 8. When x=3, u = 2(3)+8 = 6+8 = 14. Step 3: Substitute into the integral and simplify. _8^14 (u-11)/(u) · (1)/(2)\,du = (1)/(2) _8^14 (1 - (11)/(u))\,du Step 4: Integrate and evaluate using the new limits. (1)/(2) [u - 11|u|]_8^14 (1)/(2) [(14 - 11|14|) - (8 - 11|8|)] (1)/(2) [14 - 11(14) - 8 + 11(8)] (1)/(2) [6 + 11((8) - (14))] (1)/(2) [6 + 11((8)/(14))] (1)/(2) [6 + 11((4)/(7))] 3 + (11)/(2)((4)/(7)) The final answer is 3 + (11)/(2)((4)/(7)). Send me the next one 📸