Substitution Rule Applied to Indefinite Integrals

When a force is applied to a linear spring, the restoring force increases proportionally with the amount of displacement. This behavior is described by Hooke’s law, which allows the work done on the spring to be determined directly from the force–displacement relationship. In this case, the force varies in a simple and predictable manner, making the calculation relatively simple.

On the other hand, a nonlinear spring does not obey Hooke’s law. Its restoring force depends on position in a non-proportional way, causing the force–displacement relationship to become curved rather than linear. One example of such a restoring force is given by the expression

where the force increases more rapidly than the displacement alone. This form shows how nonlinear effects become more pronounced as the spring is stretched further.

The total work done in stretching a nonlinear spring is determined by accumulating the force over the displacement. However, evaluating this quantity directly is difficult because the square root term complicates standard integration techniques. To simplify the calculation, the substitution rule is applied. This method introduces a new variable to represent the expression inside the square root, making the remaining terms easier to handle.

By differentiating the substituted variable and rearranging terms, the expression can be rewritten in a simpler form that reduces to a single power. After carrying out the integration, the substituted variable is replaced with the original displacement variable so that the physical meaning of the result is preserved. The final expression represents the work required to stretch the nonlinear spring.