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
Welcome back Somma🥰🥰 — been a while! Let's pick up where you left off. To find the integral xsqrt(1-x^2)\,dx, we will use a substitution method. Step 1: Set up the substitution. Let u = 1-x^2. Step 2: Find the differential du. Differentiate u with respect to x: (du)/(dx) = -2x Rearrange to find x\,dx: du = -2x\,dx x\,dx = -(1)/(2)\,du Step 3: Substitute into the integral. Replace 1-x^2 with u and x\,dx with -(1)/(2)\,du: xsqrt(1-x^2)\,dx = sqrt(u) (-(1)/(2))\,du = -(1)/(2) u^1/2\,du Step 4: Integrate with respect to u. Use the power rule for integration, u^n\,du = u^n+1n+1 + C: -(1)/(2) u^1/2\,du = -(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 5: Substitute back x for u. Replace u with 1-x^2: -(1)/(3)(1-x^2)^3/2 + C Comparing this result with the given options, it matches option B. The final answer is B: -(1)/(3)(1-x^2)^3/2 + c. Send me the next one 📸