A force `F=a+bx` acts on a particle in x-direction, where a and b are constants. Find the work done by this force during the displacement from `x_1` to `x_2`.

Explanation

Let the particle at any instant be at a position x. Let, under the action of force, `F=a+bx`, it describes a small displacement `dx`. Work done during the displacement `dx` will be

`dW=Fdx=(a+bx)dx`
Total work done can be obtained by "summing up" the work done in individual elemental displacements (i.e., by integrating).
Then, `W=intdW`
`=underset(x_1)overset(x_2)int(a+bx)dx` [Here x varies from `x_1` to `x_2`]
`=[ax+(bx^2)/(2)]_(x_1)^(x_2)`
`=a(x_2-x_1)+b/2(x_2^2-x_1^2)`
`=(x_2-x_1)/(2)[2a+b(x_1+x_2)]`