Antiderivatives: Rules, Formulas, and Examples

An antiderivative of a function \(f(x)\) is any function \(F(x)\) whose derivative equals \(f(x)\). In other words, if \(F'(x) = f(x)\), then \(F(x)\) is called an antiderivative of \(f(x)\). Finding an antiderivative means working backward from a derivative to the original function, which is why this process is often called "anti-differentiation."

For example, since the derivative of \(x^2\) is \(2x\), we say that \(x^2\) is an antiderivative of \(2x\). But notice that \(x^2 + 5\) also has a derivative of \(2x\), and so does \(x^2 - 100\). This happens because the derivative of any constant is zero, so antiderivatives are never unique on their own.

Because adding any constant does not change a derivative, every function has infinitely many antiderivatives that all differ by a constant. We write the complete answer using a general constant \(C\):

If \(F'(x) = f(x)\), then the most general antiderivative of \(f(x)\) is \(F(x) + C\), where \(C\) is any real number.

This collection of functions is exactly what is meant by an indefinite integral, written \(\int f(x)\,dx = F(x) + C\). The symbol \(\int\) tells you to find the antiderivative, and \(dx\) shows which variable you are integrating with respect to. Antiderivatives and indefinite integrals describe the same idea: one is the function itself, and the other is the standard notation used to write it.

The figure below shows one member of the antiderivative family of \(f(x) = 2x\), namely \(F(x) = x^2\). Shifting this curve up or down by any constant \(C\) produces every other valid antiderivative of \(2x\).

The most useful basic rule for antiderivatives is the reverse of the power rule for derivatives:

\(\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C\), for any \(n \ne -1\)

To use this rule, add one to the exponent and then divide by the new exponent. Here are a few common cases:

• Antiderivative of \(x\): since \(x = x^1\), \(\int x \, dx = \dfrac{x^2}{2} + C\)
• Antiderivative of \(2x\): \(\int 2x \, dx = 2 \cdot \dfrac{x^2}{2} + C = x^2 + C\)
• Antiderivative of a constant \(k\): \(\int k \, dx = kx + C\)

Notice why the rule excludes \(n = -1\): dividing by \(n + 1\) would mean dividing by zero. The function \(\dfrac{1}{x}\) needs a different rule entirely.

Since the derivative of \(\ln|x|\) is \(\dfrac{1}{x}\), the antiderivative of \(\dfrac{1}{x}\) is:

\(\int \dfrac{1}{x} \, dx = \ln|x| + C\)

The absolute value is important because \(\ln|x|\) is defined for negative values of \(x\) as well as positive ones, matching the domain of \(\dfrac{1}{x}\).

Trig antiderivatives come directly from reversing the derivative rules for sine, cosine, and tangent. The table below summarizes the most common ones.

The antiderivative of \(\tan x\) is less obvious than the others because tangent is not simply the derivative of a basic function you likely already know. It comes from rewriting \(\tan x\) as \(\dfrac{\sin x}{\cos x}\) and using a substitution, giving \(\int \tan x \, dx = \ln|\sec x| + C\), which can also be written as \(-\ln|\cos x| + C\).

Example 1: Find the antiderivative of \(f(x) = 2x\).

Using the power rule, \(\int 2x \, dx = x^2 + C\). Check the answer by differentiating: \(\dfrac{d}{dx}(x^2 + C) = 2x\), which matches \(f(x)\), confirming the result.

Example 2: Find the antiderivative of \(f(x) = \sin x\).

From the trig table above, \(\int \sin x \, dx = -\cos x + C\). Differentiating \(-\cos x + C\) gives \(\sin x\), so the answer checks out.

Example 3: Find the antiderivative of \(f(x) = \dfrac{1}{x} + x^3\).

Antiderivatives can be found term by term: \(\int \left(\dfrac{1}{x} + x^3\right) dx = \ln|x| + \dfrac{x^4}{4} + C\).

Antiderivatives are the foundation for evaluating a definite integral, since the fundamental theorem of calculus shows how to compute a definite integral by finding an antiderivative and evaluating it at the interval's endpoints. This connects the idea of an antiderivative directly to the concept of area under a curve, which is first approximated using a Riemann sum before being calculated exactly through integration.

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