Answer:
[tex]f(x)=\frac{\sin^2x}{2}[/tex]
Explanation:
Given that the derivative of a function:
[tex]\frac{df}{dx}=\sin(x)\cos(x)[/tex]
To find the function f(x), evaluate the antiderivative (or the integral).
[tex]\int\frac{df}{dx}dx=\int\sin(x)\cos(x)dx[/tex]
Let u=sin(x)
[tex]\begin{gathered} \frac{du}{dx}=cos(x)\implies dx=\frac{du}{\cos(x)} \\ Therefore \\ \implies\int sin(x)cos(x)dx=\int u\cos(x)\frac{du}{\cos(x)}=\int udu \end{gathered}[/tex]
Apply the power rule:
[tex]\int udu=\frac{u^2}{2}[/tex]
Undo the substitution: u=sin(x)
[tex]f(x)=\frac{\sin^2x}{2}[/tex]
A correct function for f(x) is:
[tex]f(x)=\frac{\sin^2x}{2}[/tex]
