Basic Antiderivative Formulas for Indefinite Integrals

Hello! In our last lesson, we introduced the fundamental concept of the indefinite integral as the reverse process of differentiation. We established the notation $\int f (x) d x = F (x) + C$ and learned our first integration technique: the power rule.

Today, we're going to significantly expand your ability to find antiderivatives. The goal of this lesson is to learn how to evaluate indefinite integrals using a range of basic antiderivative formulae (formulas). We'll build a "table of integrals" for common functions like exponential and trigonometric functions, and we'll solidify the rules that allow us to integrate more complex expressions term by term.

1. The Rules of Engagement: Linearity of Integrals

In the last lesson, when we integrated a polynomial like $x^{2} + 3 x$ , we intuitively integrated each term separately. This intuition is based on two fundamental properties of integrals, which together are known as linearity. These are direct consequences of the corresponding rules for differentiation.

The Sum/Difference Rule: The integral of a sum (or difference) of functions is the sum (or difference) of their integrals.
$\int [f (x) \pm g (x)] d x = \int f (x) d x \pm \int g (x) d x$
The Constant Multiple Rule: A constant factor can be moved outside the integral sign.
$\int k \cdot f (x) d x = k \int f (x) d x$
(where $k$ is a constant)

These rules allow us to break down complex-looking integrals into a series of simpler ones that we can solve individually.

Basic Indefinite Integration Rules and Examples — This table summarises the basic rules for indefinite integration, including the Sum, Difference, and Constant Multiple rules, each with a clear example.

2. A Table of Basic Integrals

To make use of the rules above, we need a library of antiderivatives for common functions. Since integration is the reverse of differentiation, we can build this library by reading a table of derivatives backwards.

Below is a summary of the most important indefinite integrals you should become familiar with.

Function $f (x)$	Indefinite Integral $\int f (x) d x$	Notes
$k$ (a constant)	$k x + C$
$x^{n}$	$\frac{x ^{n + 1}}{n + 1} + C$	Power Rule (for $n \neq = - 1$ )
$x^{- 1}$ or $\frac{1}{x}$	$\ln	x
$e^{x}$	$e^{x} + C$
$e^{k x}$	$\frac{1}{k} e^{k x} + C$
$sin (x)$	$- cos (x) + C$	Note the sign change!
$cos (x)$	$sin (x) + C$
$sec^{2} (x)$	$tan (x) + C$

A more comprehensive table is an essential tool at this stage. The following resource, from a UK-based academic support centre, provides an excellent summary using the terminology you are familiar with.

HELM workbook 13 - Integration

The HELM workbook provides a concise table of standard integrals and a formal explanation of the integration rules. This will serve as our primary reference for the basic antiderivative formulae.

Please read Section 13.1, focusing on subsection 2, 'A table of integrals' (page 4), and subsection 3, 'Some rules of integration' (pages 8-9). Familiarise yourself with the standard integrals in the table and the rules for integrating sums and constant multiples.

The Special Case: Integrating $\frac{1}{x}$

Notice that the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ fails for $n = - 1$ , as it would lead to division by zero. This case corresponds to integrating $f (x) = x^{- 1} = \frac{1}{x}$ .

Recall from differentiation that $\frac{d}{d x} (ln (x)) = \frac{1}{x}$ . Reversing this gives us the integral. We use the absolute value, $ln ∣ x ∣$ , to extend the domain of the antiderivative to include negative values of $x$ , since $ln (x)$ is only defined for $x > 0$ .

3. Applying the Rules and Formulae

Now, let's put everything together. The general strategy for solving basic indefinite integrals is:

Rewrite the integrand: Use algebra and the laws of indices (exponents) to get each term into a form that matches our table of integrals (e.g., $x^{n}$ , $e^{k x}$ , etc.).
Integrate: Apply the Sum/Difference and Constant Multiple rules to integrate term by term using the formulae from the table.
Simplify: Tidy up the resulting expression, and don't forget to add the constant of integration, $+ C$ , at the end.

The following video provides an excellent walkthrough of this entire process with many worked examples.

Indefinite Integration || Reverse Power Rule, Basic Integration Rules, Integral Table | Calculus

Now let's see these rules and formulae in action. This video works through a variety of examples, from polynomials and fractions to trigonometric functions. It's a great way to see the problem-solving process.

Watch from 12:35 to the end (38:35). Pay close attention to how the presenter: Explains why integrating 1/x is a special case (around 18:30). Applies the sum and constant multiple rules (from 19:42). Simplifies expressions before integrating, like rewriting radicals and splitting fractions (around 26:30). Uses the table of integrals for trig functions (from 37:30).

Test your understanding!

Find the indefinite integral $\int (5 e^{x} + 2 cos (x) - \frac{3}{x}) d x$ .

Show answer

We can integrate term by term:
$\int (5 e^{x} + 2 cos (x) - \frac{3}{x}) d x = 5 \int e^{x} d x + 2 \int cos (x) d x - 3 \int \frac{1}{x} d x$
Using our table of formulae:
$= 5 (e^{x}) + 2 (sin (x)) - 3 (ln ∣ x ∣) + C$
$= 5 e^{x} + 2 sin (x) - 3 ln ∣ x ∣ + C$
Note that we only need a single constant of integration for the final expression.

4. Rewriting Before Integrating

As the video demonstrated, many functions need to be algebraically manipulated before they can be integrated with our basic rules. This is the most common place where mistakes are made.

Let's work through an example that requires rewriting. Find $\int \frac{x ^{3} - 4 x}{x} d x$ .

Rewrite: The integrand is a fraction. We can simplify it by dividing each term in the numerator by the denominator:
$\frac{x ^{3} - 4 x}{x} = \frac{x ^{3}}{x} - \frac{4 x ^{1/2}}{x ^{1}} = x^{2} - 4 x^{(1/2 - 1)} = x^{2} - 4 x^{- 1/2}$
Our integral is now $\int (x^{2} - 4 x^{- 1/2}) d x$ . This is in a form we can handle.
Integrate: Apply the power rule to each term:
$\int x^{2} d x - 4 \int x^{- 1/2} d x$
For the first term: $\frac{x ^{2 + 1}}{2 + 1} = \frac{x ^{3}}{3}$ .
For the second term: $- 4 (\frac{x ^{- 1/2 + 1}}{- 1/2 + 1}) = - 4 (\frac{x ^{1/2}}{1/2}) = - 4 \cdot 2 x^{1/2} = - 8 x^{1/2}$ .
Simplify and add C: Combine the terms and add the constant of integration.
$\frac{x ^{3}}{3} - 8 x^{1/2} + C = \frac{1}{3} x^{3} - 8 x + C$

A Final Tip: Always Check Your Work!
Integration can be tricky, but you have a built-in way to check your answer. Simply differentiate your result. You should get back the original integrand.
For our example above:
$\frac{d}{d x} (\frac{1}{3} x^{3} - 8 x^{1/2} + C) = \frac{1}{3} (3 x^{2}) - 8 (\frac{1}{2} x^{- 1/2}) + 0 = x^{2} - 4 x^{- 1/2}$ . This matches the rewritten integrand.

Conclusion

In this lesson, you've acquired the core tools for finding a wide range of indefinite integrals. You are no longer limited to simple polynomials.

Key Takeaways:

Indefinite integrals follow the rules of linearity: you can integrate sums/differences term by term, and pull constant multiples out of the integral.
We have a table of standard antiderivatives for power functions, exponential functions, and key trigonometric functions.
The integral of $\frac{1}{x}$ is a special case, resulting in $ln ∣ x ∣ + C$ .
A crucial first step is often to rewrite the integrand using algebra and the laws of indices to match the forms in our integral table.
You can always check your integration by differentiating the result.

So far, we have only dealt with indefinite integrals, which result in a family of functions. In our next lesson, we will begin our exploration of the definite integral. You will learn how to approximate the area under a curve using a method called Riemann sums, which forms the conceptual bridge to the second great part of the Fundamental Theorem of Calculus.

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