Lesson Explainer: Antiderivatives Mathematics

In this explainer, we will learn how to find the antiderivative of a function. The antiderivative of a function is the function where .

An antiderivative, also known as an inverse derivative or primitive, of a function is another function whose derivative is equal to the original function .

We note that since is the derivative of , is an antiderivative of ; similarly, is an antiderivative of and is an antiderivative of . The process of finding an antiderivative of a function is the reverse process of differentiating a function; for example, is the derivative of , so we can say that is an antiderivative of .

There are many real-world applications of antiderivatives, for example, when we consider the equations of motion in Newtonian mechanics. The velocity, , is defined as the rate of change of the displacement, , with respect to time, . In other words, velocity is the derivative of displacement:

This means the reverse is also true; displacement is an antiderivative of velocity and so the function is an antiderivative of the function . Similarly, acceleration, , is the derivative of velocity, which means that velocity is an antiderivative of acceleration.

In fact, an antiderivative is not unique and there are many functions, which differ up to a constant, that give the same derivative. In order to see this, let’s consider the constant function . The derivative of this function with respect to is

This is expected since the function does not vary as varies; thus its derivative is zero. This means that an antiderivative of is . This is also true for , and in fact any real constant , since the derivative will always be , as . This means an antiderivative of will be some constant ; or we can say that the antiderivative of is , for all .

What about the derivative of ? Since the derivative is linear, that is, we can differentiate linear combinations of functions separately, in particular, , we have,

The constant C, also known as the constant of antidifferentiation, is very important as it produces a family of antiderivatives, , parameterized by C. This means that is the derivative of , so that is the antiderivative or the most general antiderivative of , for all ; this is why we always add a when finding the (general) antiderivative of any function as without it we only have an antiderivative, which is not unique. In other words, is the most general function that has a derivative , for all ; for example, while is an antiderivative of , is the general antiderivative of . This means that , , , or , and so on, are all also antiderivatives of .

By using this, we can also say that the general antiderivative of a function is the function , for , and similarly the general antiderivative of the function is the function and this process continues.

If we are also given , also known as a boundary condition, then we can also determine the constant C to give a unique antiderivative that satisfies the boundary condition.

We also note that a constant multiple in front of a function does not affect its antiderivative. This follows from the definition of the derivative applied to :

Therefore, is an antiderivative of , or is the general antiderivative of , for all . This means we can always bring a constant multiple inside the derivative as ; in other words, if the derivative gets multiplied by a constant, the antiderivative also gets multiplied by the same constant and vice versa.

If we want to find the antiderivative of for some functions and constants , we can find the antiderivatives of and separately, say and , and add the results. In other words, the antiderivative of a sum is the sum of antiderivatives; the antiderivative is a linear operation, which follows from the linearity of the derivative.

In order to see this, we let , with and , and using the fact that we can differentiate linear combinations of functions separately, we have

Thus, an antiderivative of the sum is , the sum of antiderivatives. Therefore, in order to work in reverse and find an antiderivative of a sum of functions, we just find an antiderivative of each part separately and add the results together, not forgetting the at the end. Usually, we would obtain multiple constants for each part from the process of antidifferentiation, but we can combine these into one constant. The most general antiderivative of is

We note that while we have been writing the most general antiderivative as to make the constant appear explicitly, we can usually absorb this in the function so that it already contains . For example, we can say that the general antiderivative of is , instead of writing the general antiderivative as with .

Now, suppose for ; the derivative of this function can be found by the power rule for differentiation as

It will be useful to rewrite this as where we have divided it by the constant , as a constant multiple of a function does not affect the derivative or antiderivative. But what if we want to work in reverse? That is, given , we want to determine the antiderivative. This means we want to find the most general function which differentiates to give .

We have already shown that the derivative of is , for . If we let , then we have

Thus, is an antiderivative of , provided . The most general antiderivative of can be found by adding the constant of antidifferentiation as which we can verify by differentiating this expression with respect to . For the case when , that is, the antiderivative of , we note that

Thus, the general antiderivative of when is

Let’s look at a few examples where we apply these rules, starting with finding an antiderivative of a polynomial function.

Now let’s consider an example where we have to determine the constant of antidifferentiation by applying a boundary condition.

The next example involves taking an antiderivative twice, in order to determine the function from its second derivative.

The next example involves finding the unique expression of a function using its first derivative and a boundary condition.

So far, we have seen how to find the antiderivative of a function involving powers of by working backward from a derivative. Similarly, we can apply the same process to some other common functions (e.g., exponential or trigonometric) that we encounter, by first computing the derivative and working backward to find an antiderivative with the inclusion of the constant . This allows us to create a table of general antiderivatives.

These can also be verified directly by computing the derivatives of to show that they equal the corresponding . For example, if , then the derivative can be found as

For more complicated functions which are given by the sum or product of these standard functions, we can use a few rules to help us determine an antiderivative, such as the linearity of the antiderivative that we established.

In order to see this, suppose we want to find an antiderivative of

This is a sum of two different types of functions: an exponential and trigonometric function. We can determine an antiderivative of and separately and add the result. The derivative of is ; thus the antiderivative of is and similarly the antiderivative of is itself. In particular, we find which we can verify by taking the derivative.

Now, let’s look at a few more examples in order to practice and deepen our understanding. The next example involves trigonometric function and a noninteger power of .

Now, let’s look at an example where we have to determine a function by its third derivative involving a trigonometric and square root function, by finding an antiderivative three times.

If we have a product of functions , where and are the antiderivatives of and with and , we know from the product rule that

Therefore, an antiderivative of the combination is and the most general antiderivative of is

In fact, we can follow this process for any derivative rule to find a corresponding rule for the antiderivative.

Now, suppose we want to find an antiderivative of

This is a sum of two functions, each a product of two different types of functions: an exponential and trigonometric function. In order to find the antiderivative, we have to reverse the product rule by identifying and and taking the product to compute an antiderivative. By comparing the first term in with the first term in the product rule, , we can have and thus the two choices or

Only one of these choices will work for our antiderivative. We can find an antiderivative of to determine for both of these choices and then compare the derivative of the product to see which would give us the function .

The first choice would give since the antiderivative of is itself and the product :

Clearly this choice works as the right-hand side is equivalent to . For completeness, let’s also examine the second choice. An antiderivative of is , which would give us the product . By taking the derivative of this product, we have

This does not work since the second term has a different sign compared to the function . Thus, the most general antiderivative of is

We note that we could have also done this with the second term in the product rule and with , but this would give us the same result; thus, it is sufficient to check just one of the terms to find the appropriate product, which is the antiderivative that we require, and then differentiating to check if this gives back the function .

Finally, consider the function

Again this is the sum of two functions, each a product of a power of and an exponential. Therefore, upon identification of and , we obtain which we can also verify by taking the derivative

Finally, let’s consider an example where we have to reverse the product rule on a function involving a square root and exponential.

In this explainer, we determined antiderivatives of a function by essentially reversing the process of differentiation. There are more direct ways to compute an antiderivative, using something called an integral. In particular, an antiderivative of a function is equivalent to the indefinite integral of . Thus, if , then where C is also called the constant of integration. To determine indefinite integrals, there are many tools we can use, which makes finding an antiderivative easier. The fundamental theorem of calculus also provides the connection between derivatives and definite integrals, which can be interpreted as the area under the curve of within an interval.

These are beyond the scope of this explainer and will be covered in another lesson in more detail.