If f(x) = (x+1)^2 and F(1) = 2, find F(x).
I can't get the answer. Could someone explain to to me? And also, what is the difference between f(x) and F(x)??
Thank you for your help.
I assume that $\displaystyle F(x)$ is the antiderivative of $\displaystyle f(x)$.
So if $\displaystyle f(x) = (x + 1)^2 = x^2 + 2x + 1$ then
$\displaystyle F(x) = \int{x^2 + 2x + 1\,dx}$
$\displaystyle = \frac{1}{3}x^3 + x^2 + x + C$.
You know that $\displaystyle F(1) = 2$, so
$\displaystyle 2 = \frac{1}{3}(1)^3 + 1^2 + 1 + C$
$\displaystyle 2 = \frac{1}{3} + 1 + 1 + C$
$\displaystyle 2 = \frac{7}{3} + C$
$\displaystyle C = -\frac{1}{3}$.
Therefore $\displaystyle F(x) = \frac{1}{3}x^3 + x^2 + x - \frac{1}{3}$.