_ -1 ^3 ( ⁻¹ ( x x^2+1 )+ ⁻¹ ( x^2+1 x ) ) d x=

Step-by-step solution

The integral is _ -1 ³ ( ⁻¹ ( x x² + 1 ) + ⁻¹ ( x² + 1 x ) ) d x . Using the identity for inverse tangents, ⁻¹(y) + ⁻¹ ( 1 y ) = 2 when y > 0 and - 2 when y Let y = x x² + 1 . Since x² + 1 > 0 for all real x , the sign of y matches that of x . For x (0, 3] , y > 0 , so the integrand equals 2 . For x [-1, 0) , y Splitting the integral at x = 0 gives: I = _ -1 ⁰ - 2 d x + ₀³ 2 d x = - 2 (0 - (-1)) + 2 (3 - 0) = - 2 + 3 2 = . The final result is .