Chapter 7, Problem 42AP

When an object is displaced by an amount x from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of x. In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as $F(x)=-(k_{1}x+k_2{x}^2+k_3{x}^3+\cdots )$. The first term here is just Hooke’s law, which describes the force exerted by a simple spring for small displacements. For small excursions from equilibrium, we generally ignore the higher-order terms, but in some cases it may be desirable to keep the second term as well. If we model the restoring force as F = −(k₁x + k₂x²), how much work is done on an object in displacing it from x = 0 to x = x_max by an applied force −F?