If 3x + 7 x^2 - 3x + 2 , dx = m ( x - 2 x - 1 ) + n (x - 2) + c , where m, n R and c is an integration constant, then m + n =

Step-by-step solution

Let I = 3x + 7 x^2 - 3x + 2 , dx = 3x + 7 (x - 1)(x - 2) , dx Using partial fractions, let 3x + 7 (x - 1)(x - 2) = A x - 1 + B x - 2 3x + 7 = A(x - 2) + B(x - 1) Substituting x = 1 , we get 10 = -A A = -10 Substituting x = 2 , we get 13 = B I = ( -10 x - 1 + 13 x - 2 ) , dx = -10 (x - 1) + 13 (x - 2) + c The given expression is I = m ( x - 2 x - 1 ) + n (x - 2) + c Expanding the logarithmic term: I = m (x - 2) - m (x - 1) + n (x - 2) + c I = (m + n) (x - 2) - m (x - 1) + c Comparing the coefficients with our integr