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
Miss, let's knock this out. Step 1: Apply substitution. Let u = sqrt(x+2). Then u^2 = x+2, which means x = u^2 - 2. From this, x+1 = (u^2 - 2) + 1 = u^2 - 1. Differentiating u^2 = x+2 with respect to x, we get 2u\,du = dx. Step 2: Change the limits of integration. When x=2, u = sqrt(2+2) = sqrt(4) = 2. When x=7, u = sqrt(7+2) = sqrt(9) = 3. Step 3: Rewrite the integral in terms of u. _2^7 (1)/((x+1)sqrt(x+2))\,dx = _2^3 (1)/((u^2-1)u) (2u\,du) = _2^3 (2u)/(u(u^2-1))\,du = _2^3 (2)/(u^2-1)\,du Step 4: Perform partial fraction decomposition for the integrand. (2)/(u^2-1) = (2)/((u-1)(u+1)) Let (2)/((u-1)(u+1)) = (A)/(u-1) + (B)/(u+1). Multiplying by (u-1)(u+1) gives: 2 = A(u+1) + B(u-1) Set u=1: 2 = A(1+1) + B(1-1) 2 = 2A A=1. Set u=-1: 2 = A(-1+1) + B(-1-1) 2 = -2B B=-1. So, the integrand becomes: (2)/(u^2-1) = (1)/(u-1) - (1)/(u+1) Step 5: Integrate the decomposed expression. _2^3 ((1)/(u-1) - (1)/(u+1))\,du = [|u-1| - |u+1|]_2^3 Using the logarithm property a - b = (a)/(b): = [|(u-1)/(u+1)|]_2^3 Step 6: Evaluate the definite integral using the limits. [|(u-1)/(u+1)|]_2^3 = |(3-1)/(3+1)| - |(2-1)/(2+1)| = |(2)/(4)| - |(1)/(3)| = ((1)/(2)) - ((1)/(3)) Again, using the logarithm property a - b = (a)/(b): = ((1/2)/(1/3)) = ((1)/(2) × (3)/(1)) = ((3)/(2)) The final answer is (3)/(2). Send me the next one 📸