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
Hey Issa, good to see you again. Here are the solutions to the integration problems. 1. i) (x+1)sqrt(2x+x^2)\,dx Step 1: Use substitution. Let u = 2x+x^2. Then, differentiate u with respect to x: (du)/(dx) = 2 + 2x = 2(1+x) So, du = 2(1+x)\,dx, which means (1+x)\,dx = (1)/(2)\,du. Step 2: Substitute u and du into the integral. (x+1)sqrt(2x+x^2)\,dx = sqrt(u) · (1)/(2)\,du = (1)/(2) u^1/2\,du Step 3: Integrate with respect to u. (1)/(2) · u^1/2+11/2+1 + C = (1)/(2) · u^3/23/2 + C = (1)/(2) · (2)/(3) u^3/2 + C = (1)/(3) u^3/2 + C Step 4: Substitute back u = 2x+x^2. (1)/(3) (2x+x^2)^3/2 + C The final answer is (1)/(3) (2x+x^2)^3/2 + C. 1. ii) _0^1 t t\,dt Step 1: Use integration by parts, u\,dv = uv - v\,du. Let u = t and dv = t\,dt. Then, du = dt and v = t\,dt = t. Step 2: Apply the integration by parts formula. t t\,dt = t t - t\,dt = t t - t + C Step 3: Evaluate the definite integral from 0 to 1. [t t - t]_0^1 = (1 · 1 - 1) - (0 · 0 - 0) We know that 0 = 0 and 0 = 1. = ( 1 - 1) - (0 - 1) = 1 - 1 + 1 Using the definitions x = e^x - e^-x2 and x = e^x + e^-x2: 1 - 1 = e^1 - e^-12 - e^1 + e^-12 = e - e^-1 - e - e^-12 = -2e^-12 = -e^-1 So, the expression becomes: -e^-1 + 1 The final answer is 1 - e^-1. 2. i) (^2(1/x))/(x^2)\,dx Step 1: Use substitution. Let u = (1)/(x) = x^-1. Then, differentiate u with respect to x: (du)/(dx) = -1 · x^-2 = -(1)/(x^2) So, du = -(1)/(x^2)\,dx, which means (1)/(x^2)\,dx = -du. Step 2: Substitute u and du into the integral. (^2(1/x))/(x^2)\,dx = ^2(u) (-du) = - ^2(u)\,du Step 3: Integrate with respect to u. u + C Step 4: Substitute back u = 1/x. -(1/x) + C The final answer is -(1/x) + C. 2. ii) (( x))/(x)\,dx Step 1: Use substitution. Let u = x. Then, differentiate u with respect to x: (du)/(dx) = (1)/(x) So, du = (1)/(x)\,dx. Step 2: Substitute u and du into the integral. (( x))/(x)\,dx = (u)\,du Step 3: Integrate with respect to u. u + C Step 4: Substitute back u = x. -( x) + C The final answer is -( x) + C. Send me the next one 📸