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.
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Step 1: Use integration by parts. Let u = (^-1 x)^2 and dv = dx. Then, we find du and v: du = 2(^-1 x) · (d)/(dx)(^-1 x) \, dx = 2(^-1 x) ((-1)/(sqrt(1-x^2))) \, dx v = dx = x The integration by parts formula is u \, dv = uv - v \, du. (^-1 x)^2 dx = x(^-1 x)^2 - x · 2(^-1 x) ((-1)/(sqrt(1-x^2))) \, dx (^-1 x)^2 dx = x(^-1 x)^2 + 2 x ^-1 xsqrt(1-x^2) \, dx Step 2: Evaluate the new integral using integration by parts again. Consider the integral I_2 = x ^-1 xsqrt(1-x^2) \, dx. Let U = ^-1 x and dV = (x)/(sqrt(1-x^2)) \, dx. Then, we find dU and V: dU = (-1)/(sqrt(1-x^2)) \, dx To find V, integrate dV: Let w = 1-x^2, so dw = -2x \, dx, which means x \, dx = -(1)/(2) dw. V = (x)/(sqrt(1-x^2)) \, dx = (-1)/(2) dwsqrt(w) = -(1)/(2) w^-1/2 \, dw V = -(1)/(2) (w^1/21/2) = -w^1/2 = -sqrt(1-x^2) Now apply the integration by parts formula UV - V \, dU for I_2: I_2 = (^-1 x)(-sqrt(1-x^2)) - (-sqrt(1-x^2)) ((-1)/(sqrt(1-x^2))) \, dx I_2 = -sqrt(1-x^2) ^-1 x - 1 \, dx I_2 = -sqrt(1-x^2) ^-1 x - x Step 3: Substitute I_2 back into the expression from Step 1. (^-1 x)^2 dx = x(^-1 x)^2 + 2 ( -sqrt(1-x^2) ^-1 x - x ) + C (^-1 x)^2 dx = x(^-1 x)^2 - 2sqrt(1-x^2) ^-1 x - 2x + C The final answer is x(^-1 x)^2 - 2sqrt(1-x^2) ^-1 x - 2x + C.