IB Maths AA 5.5 Definitions

This page contains our IB Maths AA definitions for 5.5. By learning each one of these definitions, you will fully cover the content for IB Maths AA 'Advanced integration'.

All Topic 5 Sub-topics 5.1: Introduction To Derivatives 5.2: Applying Derivatives 5.3: Advanced Derivatives 5.4: Introduction To Integration 5.5: Advanced Integration 5.6: Kinematics (HL)5.7: Advanced limits (HL)5.8: Implicit differentiation (HL)5.9: Challenging derivatives (HL)5.10: Challenging integration (HL)5.11: Applying integration (HL)5.12: Differential equations (HL)5.13: Maclaurin series

antiderivatives

Functions whose derivative is the given integrand; any two differ by a constant because differentiation removes constant terms.

chain rule

A differentiation rule for a composite function, used in implicit differentiation so that differentiating a term in $y$ introduces a factor of $\frac{dy}{dx}$ , for example $\frac{d}{dx}\bigl(y^2\bigr)=2y\frac{dy}{dx}$ .

composites

Functions formed by applying one function to the output of another, such as '

f\left(g(x)\right)

', often requiring a chain-rule structure when integrating.

Indefinite integration

Describing an integral that represents a family of antiderivatives and therefore includes an arbitrary constant '

C

integrals

Expressions of the form '

\int f(x)\,dx

' that represent antiderivatives of a function, reversing differentiation.

integration by inspection

Recognising an antiderivative directly by reversing a known derivative pattern, often using the reverse chain rule when the integrand matches '

g'\left(x\right)f\left(g\left(x\right)\right)

integration by substitution

Replacing one expression with another to simplify a differential equation, such as using '

y=vx

' so that '

\frac{dy}{dx}=v+x\frac{dv}{dx}

Reverse integration

Describes using the chain rule in reverse to integrate expressions matching the pattern '

g'(x)f\left(g(x)\right)

', giving '

F\left(g(x)\right)+C

' where '

F'(u)=f(u)

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