Question

Last box’s but I’m still not in Part C only

Answer 1

`f(x)=(\sin^2x)/(2)`

Explanation:

Given that the derivative of a function:

`(df)/(dx)=\sin(x)\cos(x)`

To find the function f(x), evaluate the antiderivative (or the integral).

`\int(df)/(dx)dx=\int\sin(x)\cos(x)dx`

Let u=sin(x)

`\begin{gathered} (du)/(dx)=cos(x)\implies dx=(du)/(\cos(x)) \\ Therefore \\ \implies\int sin(x)cos(x)dx=\int u\cos(x)(du)/(\cos(x))=\int udu \end{gathered}`

Apply the power rule:

`\int udu=(u^2)/(2)`

Undo the substitution: u=sin(x)

`f(x)=(\sin^2x)/(2)`

A correct function for f(x) is:

`f(x)=(\sin^2x)/(2)`