Ch 3 · 14.10–14.13 Integration · O Level Additional Mathematics 4037
Topic 3 of 4 in Calculus — O Level Additional Mathematics 4037 — 55 lessons.
Introduction
Differentiation turns a quantity into its rate of change; integration turns the rate back into the quantity. It also finds areas under curves. Every rule here is a differentiation rule read backwards, so check any integral by differentiating your answer. (Syllabus 14.10–14.13.)
Key ideas — Reverse differentiation
∫ x^n dx = fracx^n+1n + 1 + c (n ≠ -1) ∫ frac1x dx = ln x + c
🧑🏫 Teacher's explanation
Think of differentiation as taking a machine apart to see how it changes at any single moment. Integration is the reverse: it is like taking all those tiny, broken-down pieces and putting them back together to find the original shape. If differentiation tells you the speed of a car at a specific second, integration helps you work backward to find the total distance traveled.
📖 Definitions
🔍 Worked example — Easy
Find ∫ (3x^2 + 4x - 5) dx
🔍 Worked example — Standard
Evaluate ∫_1^e 1/x dx
🔍 Worked example — Exam-style
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 3.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Standard integrals
The pattern: integrate as usual, then divide by a (the reverse of the chain rule).
🧑🏫 Teacher's explanation
Integration is essentially the 'undoing' of differentiation. If differentiation tells you the slope of a curve at any point, integration works backwards to find the original function that produced that slope. Think of it like a detective working backward from a crime scene (the derivative) to find the original person (the function).\n\nWhen you differentiate a function like e^3x, you multiply by the derivative of the inside (the 3). Therefore, when you integrate, you must do the opposite: you perform the standard integration rule and then divide by that same constant. This is why every formula in this lesson includes a 1/a term—it compensates for that 'inner' constant that was multiplied during differentiation.
📖 Definitions
🔍 Worked example — Exam-style
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x=1 and x=3.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — "Hence" integrals
A common question: differentiate something, then use the result to integrate. If d/dx(xln x) = ln x + 1, then ∫(ln x + 1) dx = xln x + c, so ∫ln x dx = xln x - x + c.
🧑🏫 Teacher's explanation
In calculus, integration is the 'undo' button for differentiation. When a question asks you to find an integral "hence" a previous result, it is giving you a massive shortcut. Instead of using complex rules, you are being told: "Look at the derivative we just calculated; the integral is simply the original function plus a constant."
📖 Definitions
🔍 Worked example — Easy
Given that d/dx(x^2 sin x) = 2x sin x + x^2 cos x, find ∫ (2x sin x + x^2 cos x) dx.
🔍 Worked example — Standard
Given that d/dx(1/x) = -1/x^2, find ∫ 1/x^2 dx using the 'hence' method.
🔍 Worked example — Exam-style
A curve has the equation y = 3x^2 + 4x. Given that the gradient function is f'(x) = 6x + 4, find the area bounded by the curve, the x-axis, and the lines x=1 and x=2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Definite integrals
∫_a^b f(x) dx = big[F(x)big]_a^b = F(b) - F(a)
🧑🏫 Teacher's explanation
Think of integration as 'un-doing' a process. If differentiation tells you how fast something is changing at a specific moment (the slope), integration tells you how much total change has accumulated over an interval. It is like having a speedometer reading for every second of a journey and using it to calculate the total distance traveled.
📖 Definitions
🔍 Worked example — Easy
Evaluate ∫_1^3 (3x^2 + 4x) dx
🔍 Worked example — Standard
Evaluate ∫_0^π/4 sec^2(2x) dx
🔍 Worked example — Exam-style
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 3.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Areas
🧑🏫 Teacher's explanation
Think of integration as the process of 'un-doing' a rate of change to find a total accumulation. If differentiation tells you how fast a car is moving at a single moment, integration tells you how far it has traveled over a period of time.
📖 Definitions
🔍 Worked example — Easy
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 3.
🔍 Worked example — Standard
Find the area of the region enclosed by the curve y = 4 - x^2 and the x-axis.
🔍 Worked example — Exam-style
Find the area of the region between the line y = x + 2 and the curve y = x^2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
∫ (6x^2 + frac4x - 3/x^2 )dx = 2x^3 + 4ln x + frac3x + c.
🔍 Worked example — Easy
Find the indefinite integral: ∫ (4x^3 - 2/x + 5) dx
🔍 Worked example — Exam-style
Find the area enclosed by the curve y = x^2 and the line y = x + 2.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) tfrac23x^3/2 - 1/x^2 + c. (2) tfrac12(4x - 1)^1/2 + c. (3) tanπ/4 = 1; [-tfrac12cos 2x ]_0^π/2 = 1. (4) y = e^3x - 2x + c, 4 = 1 + c: y = e^3x - 2x + 3. (5) intersections x = -2, 1: ∫_-2^1 (2 - x - x^2) dx = tfrac92.