Question

The force needed to hold a particular spring compressed an amount x from its normal length is given by f = kx + 2ax3 + bx5. how much work must be done to compress it by an amount x, starting from x = 0? (answer in terms of a, b, k, and x, as necessary.)

Answer 1

Given the expression for the force: 
$F(x)=kx +2ax^3+bx^5$
The work done to compress the spring of an amount x is equal to the integral of F(x) in dx:
$W= \int {F(x)} \, dx =\int {(kx+2ax^3+bx^5)} \, dx =k \frac{x^2}{2}+a \frac{x^4}{2}+b \frac{x^6}{6}$