Question

When variable force moves an object along straight line, we can say that force function is the derivative of work function.Given force function f(x)= (1 + cos(2x))/2, determine work function

Answer 1

Given information: Force function is the derivate of work function. Then, work function is the integral of force function.

Given force function f(x)= (1 + cos(2x))/2. Find the indefinitie integral of the given function f to find the work function:

`\begin{gathered} f(x)=(1+\cos2x)/(2) \\ \\ \int(1+\cos2x)/(2)dx \end{gathered}`

1. Rewrite the function in the integral in the way a*f:

`\int((1)/(2)*1+\cos2x)dx`

Use the next rule:

`\int a*fdx=a*\int fdx`
`=(1)/(2)\int(1+\cos2x)dx`

Use the next rules:

`\begin{gathered} \int(f+g)dx=\int fdx+\int gdx \\ \\ \int1dx=x \\ \\ \int\cos2x=(\sin2x)/(2) \end{gathered}`
`\begin{gathered} =(1)/(2)\int1dx+\int\cos2xdx \\ \\ =(1)/(2)*(x+(\sin2x)/(2)) \\ \\ =(1)/(2)x+(\sin2x)/(4) \\ \end{gathered}`

Then, the work function is:

`w(x)=(1)/(2)x+(\sin2x)/(4)`