Antiderivatives: Learn It 1

  • Understand indefinite integrals and learn how to find basic antiderivatives for functions
  • Use the rule for integrating functions raised to a power
  • Use antidifferentiation to solve simple initial-value problems

Finding the Antiderivative

At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function [latex]f[/latex], how do we find a function with the derivative [latex]f[/latex] and why would we be interested in such a function?

We answer the first part of this question by defining antiderivatives. The antiderivative of a function [latex]f[/latex] is simply a function whose derivative is [latex]f[/latex]. But why is this concept important?

Antiderivatives are essential for solving problems where you need to reverse the process of differentiation. For instance, consider rectilinear motion: If you know an object’s position function [latex]s(t)[/latex], then its velocity [latex]v(t)[/latex] is the derivative [latex]s′(t)[/latex]. If you have the acceleration [latex]a(t)[/latex] which is [latex]v′(t)[/latex], and need to find the velocity, you would look for an antiderivative of [latex]a(t)[/latex].

This need to find antiderivatives is not limited to physics; it arises in various fields and applications, prompting the development of methods to find antiderivatives for complex functions. These methods and more are covered in detail under the topic ‘Techniques of Integration’ in the second volume of this text.

The Reverse of Differentiation

At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function [latex]f[/latex], how can we find a function with derivative [latex]f[/latex]? If we can find a function [latex]F[/latex] with derivative [latex]f[/latex], we call [latex]F[/latex] an antiderivative of [latex]f[/latex].

antiderivative

A function [latex]F[/latex] is an antiderivative of the function [latex]f[/latex] if

[latex]F^{\prime}(x)=f(x)[/latex]

 

for all [latex]x[/latex] in the domain of [latex]f[/latex].

Consider the function [latex]f(x)=2x[/latex].

Knowing the power rule of differentiation, we conclude that [latex]F(x)=x^2[/latex] is an antiderivative of [latex]f[/latex] since [latex]F^{\prime}(x)=2x[/latex].

Are there any other antiderivatives of [latex]f[/latex]?

Yes; since the derivative of any constant [latex]C[/latex] is zero, [latex]x^2+C[/latex] is also an antiderivative of [latex]2x[/latex]. Therefore, [latex]x^2+5[/latex] and [latex]x^{2}-\sqrt{2}[/latex] are also antiderivatives.

Are there any others that are not of the form [latex]x^2+C[/latex] for some constant [latex]C[/latex]?

The answer is no.

From Corollary 2 of the Mean Value Theorem, we know that if [latex]F[/latex] and [latex]G[/latex] are differentiable functions such that [latex]F^{\prime}(x)=G^{\prime}(x)[/latex], then [latex]F(x)-G(x)=C[/latex] for some constant [latex]C[/latex]. This fact leads to the following important theorem.

General Form of an Antiderivative

Let [latex]F[/latex] be an antiderivative of [latex]f[/latex] over an interval [latex]I[/latex]. Then,

  1. for each constant [latex]C[/latex], the function [latex]F(x)+C[/latex] is also an antiderivative of [latex]f[/latex] over [latex]I[/latex];
  2. if [latex]G[/latex] is an antiderivative of [latex]f[/latex] over [latex]I[/latex], there is a constant [latex]C[/latex] for which [latex]G(x)=F(x)+C[/latex] over [latex]I[/latex].

In other words, the most general form of the antiderivative of [latex]f[/latex] over [latex]I[/latex] is [latex]F(x)+C[/latex].

We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.

For each of the following functions, find all antiderivatives.

  1. [latex]f(x)=3x^2[/latex]
  2. [latex]f(x)=\dfrac{1}{x}[/latex]
  3. [latex]f(x)= \cos x[/latex]
  4. [latex]f(x)=e^x[/latex]