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
To solve the integral (sqrt(x))\,dx, we will use a substitution followed by integration by parts. Step 1: Perform a substitution. Let u = sqrt(x). Then u^2 = x. Differentiating both sides with respect to u gives 2u\,du = dx. Step 2: Rewrite the integral in terms of u. Substitute u and dx into the integral: (sqrt(x))\,dx = (u) · 2u\,du = 2 u (u)\,du Step 3: Apply integration by parts. For the integral u (u)\,du, use the integration by parts formula v\,dw = vw - w\,dv. Let v = u and dw = (u)\,du. Then dv = du and w = (u)\,du = -(u). Step 4: Substitute these into the integration by parts formula. u (u)\,du = u(-(u)) - (-(u))\,du = -u(u) + (u)\,du = -u(u) + (u) Step 5: Combine the results. Now, substitute this back into the expression from Step 2: 2 u (u)\,du = 2(-u(u) + (u)) + C Step 6: Substitute back u = sqrt(x). Replace u with sqrt(x) to express the answer in terms of x: 2(-sqrt(x)(sqrt(x)) + (sqrt(x))) + C = 2(sqrt(x)) - 2sqrt(x)(sqrt(x)) + C The final answer is 2(sqrt(x)) - 2sqrt(x)(sqrt(x)) + C. Send me the next one 📸