The value of the integral ₁^2 x ~d x (x+2)(x+3) is

Step-by-step solution

Evaluate the integral: ₁^2 x d x (x+2)(x+3) Using partial fraction decomposition, we express the integrand as A x+2 + B x+3 . Multiplying through by the denominator gives x = A(x+3) + B(x+2) . Setting x = -2 yields A = -2 and x = -3 gives B = 3 . The integral becomes ₁^2 ( -2 x+2 + 3 x+3 ) d x = [-2 |x+2| + 3 |x+3| ]₁^2 = [ | (x+3)^3 (x+2)^2 | ]₁^2 . Evaluating at the limits: ( 125 16 ) - ( 64 9 ) = ( 125 9 16 64 ) = ( 1125 1024 ) . This result corresponds to option D .