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
let's solve question 8a. We need to evaluate the definite integral _1^3 (x)/(sqrt(x+1))\,dx using the substitution u^2 = x + 1. Step 1: Perform the substitution. Given the substitution u^2 = x + 1. From this, we can express x in terms of u: x = u^2 - 1 Next, differentiate u^2 = x + 1 with respect to x to find dx: 2u (du)/(dx) = 1 dx = 2u\,du Also, the term sqrt(x+1) becomes: sqrt(x+1) = sqrt(u^2) = u (assuming u > 0 for the square root) Step 2: Change the limits of integration. When x = 1: u^2 = 1 + 1 = 2 u = sqrt(2) When x = 3: u^2 = 3 + 1 = 4 u = 2 Step 3: Substitute into the integral. Now, substitute x, sqrt(x+1), and dx into the integral, along with the new limits: _1^3 (x)/(sqrt(x+1))\,dx = _sqrt(2)^2 (u^2 - 1)/(u) (2u\,du) Step 4: Simplify and integrate. _sqrt(2)^2 2(u^2 - 1)\,du _sqrt(2)^2 (2u^2 - 2)\,du Now, integrate with respect to u: [ (2u^3)/(3) - 2u ]_sqrt(2)^2 Step 5: Evaluate the definite integral using the new limits. Substitute the upper limit and subtract the substitution of the lower limit: ( (2(2)^3)/(3) - 2(2) ) - ( 2(sqrt(2))^33 - 2sqrt(2) ) ( (2(8))/(3) - 4 ) - ( 2(2sqrt(2))3 - 2sqrt(2) ) ( (16)/(3) - (12)/(3) ) - ( 4sqrt(2)3 - 6sqrt(2)3 ) (4)/(3) - ( -2sqrt(2)3 ) (4)/(3) + 2sqrt(2)3 4 + 2sqrt(2)3 The final answer is: 4 + 2sqrt(2)3 What's next?