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
Step 1: Identify the integral and choose a substitution. The integral to evaluate is: _0^1 (^-1 x)^21+x^2 \,dx Let u = ^-1 x. Step 2: Find the differential du and change the limits of integration. Differentiate u with respect to x: du = (1)/(1+x^2) \,dx Change the limits of integration: When x=0, u = ^-1(0) = 0. When x=1, u = ^-1(1) = ()/(4). Step 3: Rewrite the integral in terms of u and evaluate it. Substitute u and du into the integral, along with the new limits: _0^()/(4) u^2 \,du Now, integrate u^2 with respect to u: [ u^2+12+1 ]_0^()/(4) = [ (u^3)/(3) ]_0^()/(4) Step 4: Apply the limits of integration. Substitute the upper and lower limits into the antiderivative: (()/(4))^33 - ((0)^3)/(3) = (^3)/(4^3)3 - 0 = (^3)/(64)3 = (^3)/(64 × 3) = (^3)/(192) The final answer is (^3)/(192).