Knowing how to find antiderivatives is one of the most important techniques that we’ll be learning in our integral calculus classes. In Physics, for example, we can find the function of the velocity given the function for the object’s acceleration. Given the rate of increase or decrease from a specimen, a researcher can find the original quantity or population. We can list more applications, but yes, these are all thanks to antiderivatives.
What if we’re given the function’s derivative, $\boldsymbol$, and this time, we need to find the function’s expression? We can work backwards and find an expression for the function, $\boldsymbol$. This is the foundation of antiderivatives: $\boldsymbol$ is the antiderivative of $\boldsymbol$.
In this article, we’ll show you how derivatives and antiderivatives are related to each other. We’ll also learn about the different antiderivatives that will come in handy when finding the antiderivative of a given function. You’ll also be able to test out your understanding of the exercises that follow!
The antiderivative of a function simply the result of reversing the derivative of a given function. When we have $F(x)$, it is considered the antiderivative of $f(x)$ when $\boldsymbol
A great guess would be $F(x) = x^4$, but if we try to differentiate this using the power rule, we’ll end up with $F^<\prime>(x) = 4x^3$. To account for the coefficient, $F(x)$ must have $\dfrac$ as its constant factor. Hence, $F(x) = \dfracx^4$. Now, how do we list down three unique functions? Recall that the derivative of constant will always be zero, so we simply add a unique constant to $F(x)$:
Antiderivative
Derivative ($\boldsymbol
\beginF_1(x) &= \dfracx^4 + 1\\F_2(x) &= \dfracx^4 -1 \\F_3(x) &= \dfracx^4 + \pi \end
This shows that a function may have multiple antiderivatives since we can add any constant after the antiderivative, $F(x)$, and the derivative function, $f(x)$, will still be the same.
Here’s a helpful guide to remember how derivatives and antiderivatives are related to each other:
This means that finding the antiderivative is simply reversing the process of finding the function’s derivative.
In the previous table, we’ve shown you that there can be multiple possible antiderivatives for one given function. That’s because expressions with identical terms except for their constants will have identical derivatives.
Let’s say we have $F(x)$ and $G(x)$, where the two functions only vary by a constant $C$. This means that they’ll be sharing a common derivative, $f(x)$. Since $F(x)$ and $G(x)$ differ by $C$, we can rewrite $G(x)$ as $F(x) +C$.
This generalizes the antiderivatives of $f(x)$ as $F(x) + C$, where $C$ is an arbitrary constant.
This is the general form of a function’s antiderivative. Keep in mind that $C$ is an arbitrary constant and $\boldsymbol$ is the antiderivative of $\boldsymbol$. The process of antidifferentiation is simply finding the function’s antiderivative.
Here’s an example of a family of antiderivatives that shared the same derivative of $2x$. From this, we can see that that the functions of the form $x^2 + C$ will always have a derivative of $2x$.
Integration and antidifferentitation are interchangeable processes since we’re performing the same process for both. We can also represent antiderivatives using the integration symbol, $\int$. When $f(x)$ is a function with an antiderivative of $F(x)$, we have the relationship shown below:
\begin\int f(x) \phantomdx &= F(x) + C\end
This means that $\int f(x) \phantom$ and $F(x) +C$ both represent the antiderivative or indefinite integral of $f(x)$.
Using our earlier example, since $\dfrac \dfracx^4 = x^3$, we have $\int x^3 \phantomdx = \dfracx^4 + C$.
This is a more common format for antiderivatives, so make sure to familiarize yourself with rewriting antiderivatives using the symbol, $\int$. We call this the integral symbol. The function that’s being integrated is called the integrand.
One way for us to find the antiderivative of a function is by working backward and recalling a derivative rule that may have been applied. Given $f(x)$, we’ll need to think of a function, $F(x)$, that satisfies the condition, $F^<\prime>(x)= f(x)$.
For example, if we want to find the antiderivative of $y =6$, we’re simply looking for a function that when differentiated, returns $6$. Recall that when we have $kx$, $\dfrac kx = k$, where $k$ is a constant. This means that the antiderivative of $y = 6$ is $6x +C$.
\begin\dfrac 6x &= 6\\ \int 6\phantomdx &= 6x + C\end
We can apply a similar process to derive the different integral rules and antiderivative formulas. Below are a few examples of integral rules derived from known derivative rules.
\begin \dfrac \sin x = \cos x & \Rightarrow & \int \cos x \phantomdx = \sin x + C \\\dfrac \cos x = -\sin x & \Rightarrow & \int -\sin x \phantomdx = \cos x + C \\ \dfrac e^x = e^x & \Rightarrow & \int e^x \phantomdx = e^x + C\\\dfrac \dfrac \cdot x^ = x^n & \Rightarrow & \int x^n \phantomdx = \dfrac
When finding the integral of more complex functions, combine indefinite integral properties and antiderivative rules. For now, take a look at this comprehensive list of helpful antiderivative rules.
Here’s a list of integral or antiderivative formulas that will be handy when finding the antiderivative of a function. We’re showing the derivative rule used to come up with each integration rule shown to highlight how closely related these two concepts are.
Differentiation Rule
Integration Rule
\begin\int k \phantomdx = kx + C \end
