# Antiderivatives

Definition:

In order to calculate the value of the definite integral without using the limit definition, we’re going to need to take a detour and talk about antiderivatives. An antiderivative is exactly what it sounds like: the opposite of a derivative. If $f(x)$ is a function, we define the antiderivative, $F(x) = \int f(x) dx$ to be the function such that $F'(x) = f(x)$.

Note that an antiderivative is not unique. For example, both $x^2$ and $x^2 + 3$ are antiderivatives for $f(x) = 2x$. Why is this? Well, recall that the derivative of a constant function is 0. Therefore, you can add any constant to an antiderivative and it will still be valid. Therefore, it is conventional to add a “+C” at the end of any antiderivative to account for any valid antiderivative.

Based on this, we can provide a list of elementary functions and their antiderivatives:

You’ll notice a few things. First, that each of these is simply one of our derivative rules, stated backwards. This makes sense, seeing as we are taking antiderivatives. And second, that each of these rules has a “+C” on the end. This is important to remember: EVERY TIME that you take an antiderivative, you must add a +C to account for every possible answer.

Finally, we can carry over the sum and difference rule from derivatives:

Sum and Difference Rule: Given two functions, $f(x)$ and $g(x)$, the antiderivative of $f(x) + g(x)$ is $\int f(x) dx + \int g(x) dx$. Similarly, the derivative of  $f(x) - g(x)$ is $\int f(x) dx - \int g(x) dx$.

Notice that the product, quotient, and chain rules do NOT carry over. Their inverses are a little more complicated, and we’ll discuss them later.

Example: Find the antiderivative of $f(x) = x^3 + \sin(x) - e^x$. We use the sum and difference rule and our derivative chart:

$F(x) = \int f(x) dx = \frac{x^4}{4} - \cos(x) - e^x + C$

Notice that we only put one +C at the end. We didn’t need one for each function, since each constant is arbitrary, and since each +C is just a number, we can make our equation simpler by adding them all up, putting them at the end, and calling them just one thing.

Antiderivatives are also sometimes called indefinite integrals.

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