Further Integration Key Terms: A Comprehensive Glossary

Introduction

Calculus has its own strict language, and Extension 2 pushes it to the limit. In HSC Mathematics Extension 2, integrating complex functions requires advanced algebraic teardowns before you even write an integral sign. This glossary provides the definitive mathematical definitions and illustrative examples for the core terminology used throughout the Further Integration syllabus.

Executive Summary

This guide serves as a quick-reference dictionary for:

• Core Calculus Terms: Definite/Indefinite integrals, integrands, functions.
• Algebraic Preparation: Completing the square, partial fractions, identity, substitution.
• Polynomial Properties: Degree, linear, quadratic, denominator, distinct.
• Advanced Techniques: Integration by parts, recurrence, t-formulas.

What is this about?

When an exam question says "Use the method of partial fractions...", you must know exactly what structure to apply. Understanding the difference between a "distinct linear factor" and a "quadratic expression" changes the entire algebra of the question. This page ensures you know exactly which integration technique corresponds to which mathematical term.

Main Content: Key Terms

C–D

• Completing the Square: An algebraic technique used to rewrite a quadratic expression $ax^2 + bx + c$ into the form $a(x-h)^2 + k$. In integration, this is used to force a denominator into an inverse tangent ($\tan^{-1}$) format.
  • Example: $x^2 + 6x + 13 = (x^2 + 6x + 9) + 4 = (x+3)^2 + 2^2$.
• Definite Integral: An integral with upper and lower limits that evaluates to a specific numerical value (often representing area or volume), rather than a function. It does not require a $+ C$.
  • Example: $\int_0^1 2x \, dx = [x^2]_0^1 = 1$.
• Degree (Polynomial): The highest exponent (power) of the variable in a polynomial expression.
  • Example: The degree of $P(x) = 4x^3 - 2x + 1$ is $3$.
• Denominator: The bottom part of an algebraic fraction.
  • Example: In the rational function $\frac{5x}{x^2 - 1}$, the denominator is $x^2 - 1$.
• Distinct: Unique; not repeating. In partial fractions, distinct linear factors mean the roots of the denominator are all different.
  • Example: $(x-1)(x+3)$ are distinct factors. $(x-1)^2$ are repeated (not distinct) factors.

F–L

• Function: A mathematical relationship where every input $x$ is mapped to exactly one output $y$, denoted by $f(x)$.
  • Example: $f(x) = e^x \sin x$.
• Identity: An equation that is always true for every valid value of its variables, regardless of what you substitute into it. (Unlike a normal equation which is only true for specific roots).
  • Example: $\sin^2 x + \cos^2 x \equiv 1$ is an identity.
• Indefinite Integral: An integral without upper and lower bounds. It evaluates to a general family of functions and MUST include a constant of integration ($+ C$).
  • Example: $\int 2x \, dx = x^2 + C$.
• Integrand: The mathematical expression or function that is currently inside the integral sign, waiting to be integrated.
  • Example: In $\int e^{2x} \, dx$, the integrand is $e^{2x}$.
• Integration: The calculus process of finding the antiderivative of a function, or calculating the continuous area under a curve.
• Integration by Parts: An advanced technique derived from the Product Rule, used to integrate the product of two different types of functions. Formula: $\int u \, dv = uv - \int v \, du$.
  • Example: Used to integrate $x \ln x$ or $e^x \cos x$.
• Linear: An algebraic expression of the first degree (highest power is 1). It graphs as a straight line.
  • Example: $3x + 5$ is a linear expression.

P–T

• Partial Fractions: An algebraic technique used to break a complex rational function (with a factorized denominator) into a sum of simpler, easily integrable fractions.
  • Example: Splitting $\frac{2}{x^2 - 1}$ into $\frac{1}{x-1} - \frac{1}{x+1}$.
• Proof: A rigorous logical argument demonstrating that a mathematical theorem or identity is invariably true.
  • Example: Proving that $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$.
• Quadratic Expression: An algebraic polynomial of the second degree (highest power is 2).
  • Example: $x^2 - 4x + 4$.
• Recurrence: A relation that defines a mathematical term based on the preceding terms. In integration, a reduction formula that defines $I_n$ in terms of $I_{n-1}$ or $I_{n-2}$.
  • Example: $I_n = \frac{1}{n} - I_{n-2}$.
• Substitution: (Integration by Substitution). A technique that reverses the Chain Rule by replacing a complex inner function with a single variable $u$, transforming the integral and the $dx$ differential.
  • Example: Letting $u = x^2 + 1$ to solve $\int 2x(x^2+1)^4 \, dx$.
• t-formulas: A set of algebraic identities based on the universal substitution $t = \tan(\frac{x}{2})$, allowing trigonometric functions to be written as rational algebraic fractions.
  • Example: Substituting $\sin x = \frac{2t}{1+t^2}$ to solve trigonometric integrals.

mini-FAQ page

Q: Is there a difference between an identity and an equation? A: Yes! $x^2 - 4 = 0$ is an equation (it's only true when $x = 2$ or $-2$). However, $x^2 - 4 = (x-2)(x+2)$ is an identity (it is true for literally any number you plug in). When finding partial fraction constants $A$ and $B$, you are using an identity, which is why you can substitute any random number for $x$ to solve it!

Q: Do I lose marks if I forget the $+ C$ on an indefinite integral? A: Yes, absolutely. It is a mathematical requirement because the derivative of any constant is zero, meaning an infinite family of curves share the same derivative.

Common mistakes to avoid

• Applying partial fractions to improper fractions: If the degree of the numerator is equal to or greater than the denominator (e.g. $\frac{x^2}{x^2+1}$), you CANNOT use partial fractions yet. You must use polynomial long division first.
• Forgetting the $du$ in substitution: Writing $\int u^2 dx$ is mathematically meaningless. You must calculate $dx$ in terms of $du$ and substitute it so the entire integral is with respect to the new variable.

Practice on Vu's Maths Hub

Fluency in terminology translates to speed in exams.

• Practice applying the correct technique to mixed integrals in the HSC Integration Booklet.
• Test your integration by parts skills at Vu's Maths Hub.

Further Readings

• Want to see these terms in action? Read our guide on Integrating Rational Functions.
• Explore more HSC math resources and full worked solutions at Vu's Maths Hub.

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