How to Compute Indefinite Integrals: Core Formulas, Antiderivatives, and Common Examples

The result of an indefinite integral must be differentiable back to the original function. The constant $C$ at the end must never be omitted—because any two antiderivatives of a function differ only by a constant.

I always differentiate my answer afterward. Only if it recovers the original integrand can the computation be considered reliable.

In the previous article, we explored the fundamental concepts of integration—including what integration is, its geometric interpretation, and its physical meaning. In this tutorial, we delve into methods for computing indefinite integrals. Indefinite integration is a foundational topic in calculus; mastering its computational techniques forms the essential groundwork for later learning of integration rules and substitution methods.

What Is an Indefinite Integral?

An indefinite integral refers to the process of integrating a function to obtain all its antiderivatives—the entire family of functions whose derivatives equal the original function. Given a function $f(x)$, its indefinite integral is conventionally written as:

$$F(x) = \int f(x) \, dx$$

Here, $F(x)$ is an antiderivative of $f(x)$, meaning $F'(x) = f(x)$. Since infinitely many such functions exist—differing only by an additive constant—we include an arbitrary constant $C$ to represent the full set. Thus, the general form of the indefinite integral is:

$$\int f(x) \, dx = F(x) + C$$

Methods for Computing Indefinite Integrals

To compute indefinite integrals, we rely on basic integration formulas and standard techniques. Below, we introduce several common methods, each illustrated with examples.

1. Constant Multiple Rule

If $c$ is a constant and $f(x)$ is integrable, then:

$$\int c f(x) \, dx = c \int f(x) \, dx$$

Example:
Compute $\int 4x^2 \, dx$:

$$\int 4x^2 \, dx = 4 \int x^2 \, dx = 4 \cdot \frac{x^3}{3} + C = \frac{4}{3} x^3 + C$$

2. Power Rule for Integration

For a power function $f(x) = x^n$, where $n \neq -1$, the indefinite integral is:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

Example:
Compute $\int x^5 \, dx$:

$$\int x^5 \, dx = \frac{x^{6}}{6} + C$$

3. Exponential Function Rule

For the exponential function $f(x) = e^x$, the indefinite integral is:

$$\int e^x \, dx = e^x + C$$

Example:
Compute $\int e^{2x} \, dx$:
Use substitution: let $u = 2x$, so $du = 2\,dx \Rightarrow dx = \frac{du}{2}$:

$$\int e^{2x} \, dx = \int e^u \cdot \frac{du}{2} = \frac{1}{2} e^u + C = \frac{1}{2} e^{2x} + C$$

4. Trigonometric Function Rules

For common trigonometric functions, we have the following standard integrals:

When computing an indefinite integral, first identify the structure of the integrand, then select the appropriate method—basic formula, substitution, or integration by parts—and always remember to add the constant of integration.

• $\int \sin(x) \, dx = -\cos(x) + C$
• $\int \cos(x) \, dx = \sin(x) + C$

Example:
Compute $\int \sin(x) \, dx$:

$$\int \sin(x) \, dx = -\cos(x) + C$$

Advanced Techniques for Indefinite Integration

Familiarity with certain advanced techniques greatly simplifies indefinite integration problems. Key examples include:

1. Integration by Parts

This technique applies to integrals involving products of two functions. Its formula is:

While reading “How to Compute Indefinite Integrals: Core Formulas, Antiderivatives, and Common Examples”, treat each illustration as a navigational card: first grasp the overall workflow, then examine why each step is justified, and finally verify boundary conditions.

$$\int u \, dv = uv - \int v \, du$$

Example:
Compute $\int x e^x \, dx$. Let $u = x$, $dv = e^x \, dx$:

Then $du = dx$, $v = e^x$, so:

$$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$$

2. Substitution Method (u-substitution)

Substitution is a powerful tool for handling complex integrands—especially when a change of variable simplifies the expression. Typically, we set $u = g(x)$, rewrite the integral in terms of $u$, and integrate accordingly.

If you haven’t fully internalized “How to Compute Indefinite Integrals: Core Formulas, Antiderivatives, and Common Examples”, revisit this card’s four-step action plan to retrace your learning path.

When reviewing “How to Compute Indefinite Integrals: Core Formulas, Antiderivatives, and Common Examples”, avoid tackling large projects all at once. Instead, test your understanding using one simple example to confirm whether the core logic is clear.

Conclusion

Computing indefinite integrals is a fundamental skill in calculus. Once you master standard integration rules and key techniques, indefinite integration becomes significantly more straightforward and efficient. Proficiency in indefinite integration builds confidence for tackling more complex integrals—and lays the essential foundation for upcoming topics like core integration rules and substitution methods.

Next, in our follow-up tutorial, we’ll explore “Core Integration Rules and Substitution Methods”—stay tuned!