Given that the conservative force is given by [tex]F=-3.0x-5.0x^2[/tex], the potential energy, U, is given by:
[tex]F= -\frac{dU}{dx} \\ \\ \Rightarrow dU=-Fdx \\ \\ \Rightarrow U= -\int\limits^b_a {F} \, dx [/tex]
Given that the potential energy associated with the force is zero when the object is at x = 0, the potential energy of the system associated with the force when the object is at x = 5.0 m is given by:
[tex]U=- \int\limits^5_0 {\left(-3.0x-5.0x^2\right)} \, dx \\ \\ =-\left[-1.5x^2- \frac{5}{3} x^3\right]^5_0=-\left(-1.5(5)^2- \frac{5}{3} (5)^3\right) \\ \\ =-(-1.5(25)- \frac{5}{3} (125))=-(-37.5-208.33) \\ \\ =-(-245.83)=\bold{245.8J}[/tex]
