Description

INTEGRATION — Complete Study Guide (009)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from basic antiderivatives to volumes of revolution — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Integration

Definition as the reverse of differentiation

Relationship diagram: differentiation ↔ integration (inverse operations)

Real-life applications (physics, engineering, economics)

Visual concept: area under a curve

2. Indefinite Integration

Notation explained: integral sign, integrand, dx, antiderivative, constant C

Power Rule — including the special case ∫(1/x)dx = ln|x|

Three worked examples: basic integration, negative powers, and fractional powers

3. Trigonometric Integrals

Complete formula table (sin, cos, sec², cosec², sec·tan, cosec·cot)

Worked examples including exponential and logarithmic integrals

4. Integration by Substitution

5-step method

5 fully worked examples:

Polynomial: (2x + 1)⁵

Trigonometric: sin x · cos x

Exponential: 2x·e^(x²)

Rational: x/(x² + 1)

Square root: x/√(1 − x²)

5. Integration by Parts

Formula and LIATE rule for choosing u

Table showing which function type to choose for u and dv

Three fully worked examples: x sin x, x·e^x, and ln x

6. Trigonometric Integrals with Identities

Even powers: sin²x and cos²x (using double-angle identities)

Odd powers: sin³x (rewrite and substitute)

Products: sin x cos x (two methods — substitution and identity)

7. Definite Integration

Fundamental theorem formula: F(b) − F(a)

Key properties including reversing limits

Worked examples: evaluating definite integrals and applying substitution with changed limits

8. Area Under a Curve

Formula for regions above and below the x-axis

Three worked examples:

Simple area under a parabola

Area where the curve is above the axis

Sine curve requiring splitting into positive and negative regions

9. Area Between Two Curves

Formula: ∫[f(x) − g(x)]dx

Three fully worked examples:

Parabola and line

Two parabolas (finding intersection points)

Trigonometric curves (sin x and cos x) requiring splitting at the intersection

10. Area Using Parametric Equations

Formulas for parametric area

Fully worked example

11. Volume of Revolution

Disk method (about x-axis and y-axis)

Washer method (region between two curves)

Cylindrical shell method

Three fully worked examples for each method

12. Chapter Summary

Key integration formulas table

Integration methods table (when to use substitution, parts, identities, partial fractions)

Applications summary table (area, volume formulas)

13. Quick Reference Card

Power rule, special case, substitution, parts, and definite integral on one card

14. Practice Questions

Six sections: Indefinite Integration, Substitution, Integration by Parts, Definite Integration, Area Under Curves, and Volume of Revolution

✅ Why This Guide Works
Every technique demonstrated — substitution, parts, trig identities, and all three volume methods

Multiple approaches shown — e.g., sin x cos x solved two ways

Sign handling made clear — splitting integrals for regions below the x-axis

Exam-focused — method selection tables, applications summary, and practice questions by topic

Complete coverage — from the definition of an antiderivative to volumes of revolution in a single document

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