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:

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

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

$\int(\frac{1}{2}*1+\cos2x)dx$

Use the next rule:

$\int a*fdx=a*\int fdx$$=\frac{1}{2}\int(1+\cos2x)dx$

Use the next rules:

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

Then, the work function is:

$w(x)=\frac{1}{2}x+\frac{\sin2x}{4}$