We would like to know the velocity of the block when it reaches some position x. finding this requires an integration. however, acceleration is defined as a derivative with respect to time, which leads to integrals with respect to time, but the force is given as a function of position. to get around this, use the chain rule to find an alternative definition for the acceleration ax that can be written in terms of vx and dvxdx. this is a purely mathematical exercise; it has nothing to do with the forces given in the problem statement.