x (x-1)(x-2)^2 d x=a | x-1 x-2 |+ b (x-2) +c then

Step-by-step solution

Let the integrand be x (x-1)(x-2)^2 = A x-1 + B x-2 + C (x-2)^2 . Multiplying by (x-1)(x-2)^2 , we get x = A(x-2)^2 + B(x-1)(x-2) + C(x-1) . Setting x=1 , we get 1 = A(1-2)^2 , so A=1 . Setting x=2 , we get 2 = C(2-1) , so C=2 . Comparing the coefficients of x^2 , we get 0 = A + B , so B = -A = -1 . Thus, the integral is ( 1 x-1 - 1 x-2 + 2 (x-2)^2 ) dx . Evaluating the integral, we get |x-1| - |x-2| - 2 x-2 + c = | x-1 x-2 | - 2 x-2 + c . Comparing this with the given form a | x-1 x-2 | + b x-2 + c , we identify a