SL 5.10—Indefinite integration, reverse chain, by substitution: IB DP Math AA Notes

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.

What is Indefinite Integration?

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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What is Indefinite Integration?

Standard Indefinite Integrals to Memorise

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