**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 is a function, we define the antiderivative, to be the function such that .

Note that an antiderivative is not unique. For example, both and are antiderivatives for . 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, and , the antiderivative of is . Similarly, the derivative of is .

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 . We use the sum and difference rule and our derivative chart:

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*.

Still need help with finding antiderivatives? Download Yup and get help from an expert math tutor 24/7.