Let I= ⁻¹ ( 2 x 1-x^2 ) d x , then I-2 x ⁻¹ x=

Step-by-step solution

Start with the integral I = ⁻¹ ( 2x 1 - x^2 ) dx . Using the identity 2 ⁻¹ x = ⁻¹ ( 2x 1 - x^2 ) , valid for -1 Apply integration by parts, letting u = ⁻¹ x and dv = 2 dx , so du = 1 1 + x^2 dx and v = 2x . This gives: I = 2x ⁻¹ x - 2x 1 + x^2 dx To evaluate the remaining integral, let t = 1 + x^2 and dt = 2x dx , which yields: 2x 1 + x^2 dx = dt t = |t| + C = (1 + x^2) + C Substitute back to find: I = 2x ⁻¹ x - (1 + x^2) + C The desired expression simplifies as: I - 2x ⁻¹ x = - (1 + x^2) + C This corresponds to op