If (x^4+1 ) x (x^2+1 )^2 dx = A |x|+ B 1+x^2 + c , then A - B is (where c is the constant of integration)

Step-by-step solution

Let I = x^4+1 x(x^2+1)^2 , dx . Rewriting the numerator as x^4+1 = (x^2+1)^2 - 2x^2 allows the integrand to be expressed as: (x^2+1)^2 - 2x^2 x(x^2+1)^2 = 1 x - 2x (x^2+1)^2 . Integrating term by term yields: I = 1 x , dx - 2x (x^2+1)^2 , dx = |x| - 2x (x^2+1)^2 , dx . Using the substitution u = x^2+1 where du = 2x , dx , the second term becomes: 1 u^2 , du = - 1 u + C = - 1 x^2+1 + C . Thus, the integral is: I = |x| + 1 x^2+1 + c . Comparing with the form A |x| + B 1+x^2 + c , we identify A = 1 and B = 1 . Therefo