If d x x^4+5 x^2+4 = A ⁻¹ x+ B ⁻¹ x 2 + c where c is a constant of integration, then

Step-by-step solution

Evaluate the integral d x x^4+5x^2+4 by factoring the denominator as (x^2+1)(x^2+4) . Apply partial fraction decomposition: 1 (x^2+1)(x^2+4) = P x^2+1 + Q x^2+4 . Multiply both sides by the denominator: 1 = P(x^2+4) + Q(x^2+1) . Equate coefficients of like terms: P + Q = 0 and 4P + Q = 1 , yielding P = 1 3 , Q = - 1 3 . Now integrate term by term: 1 3 1 x^2+1 , d x - 1 3 1 x^2+4 , d x = 1 3 ⁻¹ x - 1 6 ⁻¹ x 2 + c . Compare with the form A ⁻¹ x + B ⁻¹ x 2 + c to find A = 1 3 , B = - 1 6 . The correct option is B.