If $$egin{align*} \int_a^b f(t) dt = g(x), \end{align*}$$ which of the following is a primitive of $f(x)g(x)$? A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2022 - Paper 1

To find a primitive of the product $f(x)g(x)$, we apply the integration formula for products. Given that the integral of $f(t)$ with respect to $t$ yields $g(x)$, we differentiate this to find that:

rac{d}{dx} g(x) = f(x).

Now, we need a primitive of $f(x) g(x)$, which can be solved using the formula for the integral of a product. This leads us to:

ext{Primitive of } f(x)g(x) = rac{1}{2}[g(x)]^2 + C,

where $C$ is the constant of integration. The correct answer corresponds to option C: $\frac{1}{2}[g(x)]^2$.