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 evaluations of the integrals: 1. For sqrt(7x-1)\,dx: Step 1: Let u = 7x-1. Then du = 7\,dx, which means dx = (1)/(7)\,du. sqrt(u) · (1)/(7)\,du = (1)/(7) u^1/2\,du Step 2: Integrate u^1/2. (1)/(7) · u^3/23/2 + C = (1)/(7) · (2)/(3) u^3/2 + C = (2)/(21) u^3/2 + C Step 3: Substitute back u = 7x-1. (2)/(21) (7x-1)^3/2 + C 2. For (4 x + 7)\,dx: Step 1: Let u = 4 x + 7. Then du = 4\,dx, which means dx = (1)/(4)\,du. (u) · (1)/(4)\,du = (1)/(4) (u)\,du Step 2: Integrate (u). (1)/(4) (-(u)) + C = -(1)/(4) (u) + C Step 3: Substitute back u = 4 x + 7. -(1)/(4) (4 x + 7) + C 3. For sqrt( x) x\,dx: Step 1: Let u = x. Then du = x\,dx. sqrt(u)\,du = u^1/2\,du Step 2: Integrate u^1/2. u^3/23/2 + C = (2)/(3) u^3/2 + C Step 3: Substitute back u = x. (2)/(3) ( x)^3/2 + C 4. For x x\,dx: Step 1: Use integration by parts, v\,dw = vw - w\,dv. Let v = x and dw = x\,dx. Step 2: Differentiate v and integrate dw. So, dv = dx and w = x. x x\,dx = x x - x\,dx Step 3: Integrate x. x x - (- x) + C = x x + x + C x x + x + C 5. For e^x x\,dx: Step 1: Use integration by parts. Let I = e^x x\,dx. Let u = x and dv = e^x\,dx. Step 2: Differentiate u and integrate dv. So, du = x\,dx and v = e^x. I = e^x x - e^x x\,dx Step 3: Apply integration by parts again to e^x x\,dx. Let u = x and dv = e^x\,dx. Step 4: Differentiate u and integrate dv. So, du = - x\,dx and v = e^x. e^x x\,dx = e^x x - e^x (- x)\,dx = e^x x + e^x x\,dx Step 5: Substitute this back into the expression for I. I = e^x x - (e^x x + e^x x\,dx) I = e^x x - e^x x - I Step 6: Solve for I. 2I = e^x x - e^x x I = (e^x)/(2) ( x - x) + C (e^x)/(2) ( x - x) + C Drop the next question! 📸