IB Maths AI 5.5 Definitions

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

All Topic 5 Sub-topics 5.1: Introduction To Derivatives 5.2: Applying Derivatives 5.3: Fundamentals Of Integration (HL)5.4: Advanced derivatives (HL)5.5: Further integration (HL)5.6: Applying integration (HL)5.7: Kinematics (HL)5.8: Introduction to differential equations (HL)5.9: Advanced differential equations

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)

Next Up

You have completed the topic 5 definitions for IB Maths AI - continue with related resources below or explore the full IB Maths AI course from the IBO.

Explored IB Maths AI Definitions?

Get stuck into one of our other subjects!

Other subjects' Definitions

Biology

Chemistry

Physics

Maths AA

Join 85,000 students, across 130+ countries, in 500+ IB schools. That's half of the IB science graduates worldwide.

Start a 7d free trial