The integration of the function f(x) = (x^3 + 3x + 12) / (x(x+2)^2) requires the application of partial fraction decomposition. The polynomial should first be rewritten using long division to ensure the degree of the numerator is less than that of the denominator. The expression can be decomposed into the form f(x) = A/x + B/(x+2) + C/(x+2)^2, where A, B, and C are constants that can be determined by solving a system of equations. This method is essential for correctly integrating rational functions.
PREREQUISITESStudents studying calculus, particularly those focusing on integration techniques, as well as educators seeking to reinforce concepts of polynomial division and partial fractions.