Evaluate the following: 1. ∫√7x−1 dx 2. ∫sin(4πx + 7) dx 3. ∫√sin x cos x dx 4. ∫x cos x dx 5. ∫e^x sin x dx

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. Step 2: Substitute u into the integral. sqrt(u) · (1)/(7)\,du = (1)/(7) u^1/2\,du Step 3: 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 4: 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. Step 2: Substitute u into the integral. (u) · (1)/(4)\,du = (1)/(4) (u)\,du Step 3: Integrate (u). (1)/(4) (-(u)) + C = -(1)/(4) (u) + C Step 4: 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. Step 2: Substitute u into the integral. sqrt(u)\,du = u^1/2\,du Step 3: Integrate u^1/2. u^3/23/2 + C = (2)/(3) u^3/2 + C Step 4: Substitute back u = x. (2)/(3) ( x)^3/2 + C 4. For x x\,dx: Step 1: Use integration by parts, u\,dv = uv - v\,du. Let u = x and dv = x\,dx. Step 2: Find du and v. du = dx and v = x\,dx = x. Step 3: Apply the integration by parts formula. x x\,dx = x x - x\,dx Step 4: Integrate x\,dx. x x - (- x) + C = x x + x + C x x + x + C 5. For e^x x\,dx: Step 1: Use integration by parts, u\,dv = uv - v\,du. Let I = e^x x\,dx. Step 2: First application of integration by parts: Let u = x and dv = e^x\,dx. Then du = x\,dx and v = e^x. I = e^x x - e^x x\,dx Step 3: Second application of integration by parts for e^x x\,dx: Let u = x and dv = e^x\,dx. Then 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 4: Substitute this back into the equation for I. I = e^x x - (e^x x + I) Step 5: Solve for I. I = e^x x - e^x x - I 2I = e^x x - e^x x I = (1)/(2) (e^x x - e^x x) + C (e^x)/(2) ( x - x) + C Send me the next one 📸