### Home > APCALC > Chapter 3 > Lesson 3.4.2 > Problem3-166

3-166.

What is the general antiderivative, $F$, for each function below? Test your solution by verifying that $F^\prime (x) = f(x)$.

An antiderivative of a function $f$ is a function $F$ whose derivative is $f$. That is $\frac { d } { d x }F(x) = f(x)$ .

The antiderivative of $f^\prime$is $f$. However, for the antiderivative of a function $f$, we use a capital letter $F$. For example, we write the antiderivative of $g$as $G$.

Since there are an infinite number of antiderivatives that are different only by a constant term, called a “family,” we add a constant “$C$” to represent all of them. This is known as the general antiderivative (or simply the antiderivative).

For example: If $F(x) = 5x^3 + 3x^2 + C$, then $F(x) = 5x^3 + 3x^2 + 5$ and $F(x) = 5x^3 + 3x^2 − 9$ are both antiderivatives of $f$This is why we write $F(x) = 5x^3 + 3x^2 + C$.

1. $f(x) = 3x^{1/2} − 7x$

Don't forget the vertical shift $(+C)$.

1. $f(x) = \cos(x) + 2\sin(x)$

Be careful of your signs! Don't forget the vertical shift $(+C)$.