What is Indefinite Integration?

Integration is the reverse process of differentiation. Given a function $f (x)$ , its indefinite integral finds a function $F (x)$ such that $F^{'} (x) = f (x)$ .

Indefinite Integral: The indefinite integral of $f (x)$ with respect to $x$ is written $\int f (x) d x = F (x) + C$ , where $F^{'} (x) = f (x)$ and $C$ is an arbitrary constant of integration.

The word indefinite reminds us that without extra information (like a point the curve passes through), we cannot determine the exact value of $C$ , so the result is a family of functions, each one a vertical shift of the others.

Analogy

Think of differentiation as "unwrapping" a function, and integration as "re-wrapping" it. But when you re-wrap a gift, you don't know how much extra tape someone used before , that unknown extra is $C$ .

Warning

Never omit $+ C$ in an indefinite integral. In IB exams, leaving it out will cost you a mark.

Standard Indefinite Integrals to Memorise

The following results form the foundation of all integration work at SL. You should be able to recall and apply these instantly.

Function $f (x)$	Integral $\int f (x) d x$	Condition
$x^{n}$	$\frac{x ^{n + 1}}{n + 1} + C$	$n \neq = - 1$
$sin x$	$- cos x + C$	,
$cos x$	$sin x + C$	,
$\frac{1}{x}$	$\ln	x
$e^{x}$	$e^{x} + C$	,

Exam Tip

Each of these can be verified by differentiation: if you differentiate the right-hand side, you should get back $f (x)$ . Use this to check your answers.

Note

The absolute value in $ln ∣ x ∣$ matters because $ln$ is only defined for positive inputs, but $\frac{1}{x}$ can be integrated for negative $x$ too. In most SL problems, $x > 0$ is implied, but writing $ln ∣ x ∣$ is technically correct and expected.

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Function

f (x)

Integral

\int f (x) d x

Condition

x^{n}

\frac{x ^{n + 1}}{n + 1} + C

n \neq = - 1

sin x

- cos x + C

cos x

sin x + C

\frac{1}{x}

$\ln

e^{x}

e^{x} + C

SL 5.10—Indefinite integration, reverse chain, by substitution

What is Indefinite Integration?

Standard Indefinite Integrals to Memorise

9 more sections in this topic

SL 5.10—Indefinite integration, reverse chain, by substitution

What is Indefinite Integration?

Standard Indefinite Integrals to Memorise

9 more sections in this topic