AHL 5.11—Indefinite integration, reverse chain, by substitution: IB DP Math AI Notes

What is Indefinite Integration?

Indefinite integration is the reverse process of differentiation. Given a function $f (x)$ , we seek an antiderivative $F (x)$ 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 called the constant of integration.

Because the derivative of any constant is zero, there are infinitely many antiderivatives of a given function — they all differ by a constant. This is why we always append $+ C$ to an indefinite integral.

Analogy

Think of differentiation as "zooming in" on a function to find its slope, and integration as "zooming out" — reconstructing the original landscape from slope information. Just as many hills can have the same slope profile if they start at different heights, many functions share the same derivative, differing only by a vertical shift ( $+ C$ ).

Standard Indefinite Integrals

The following standard results form the foundation of all integration work at HL. These should be memorised — they can all be verified by differentiating the right-hand side.

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

Warning

The rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ fails when $n = - 1$ , because you would be dividing by zero. Instead, $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ . The absolute value is essential — $ln x$ is only defined for $x > 0$ , but $\frac{1}{x}$ is defined for all $x \neq = 0$ .

Exam Tip

To verify any integration result, differentiate your answer. If you recover the original integrand, you are correct. This is always a valid check strategy in an exam.

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What is Indefinite Integration?

Standard Indefinite Integrals

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