"Integrals in calculus" is searched by students who have reached the second half of the course, after derivatives, and found that the idea is new rather than a continuation; by students revising for a board or AP exam where integration carries a quarter of the marks; and by parents trying to understand what the child is stuck on. This page explains what an integral is in both of its meanings, the two kinds, the theorem that makes them computable, the rules a first course uses, the two techniques, four worked examples, where the topic sits in each course, and the mistakes that cost marks. The printable sheet has the rules, the techniques and ten practice questions with answers.
Download the integration rules sheet (PDF) (every rule from the first course, substitution and parts with an example each, the six mistakes, and ten practice questions with answers; two pages, free).
What an integral is
As an area. Draw a curve y = f(x). The definite integral from a to b, written ∫ₐᵇ f(x) dx, is the area between the curve and the x-axis from x = a to x = b, with area below the axis counted as negative. The symbol ∫ is a stretched S for "sum": the area is imagined as a sum of thin rectangles of height f(x) and width dx, with the width shrinking to nothing.
As a total built up from a rate. If f(x) is a rate (speed in metres per second, water flowing in litres per minute, a population's growth rate per year), then the integral over an interval is the total: the distance travelled, the litres collected, the population added. This is why integrals appear in physics, economics and biology: anything that accumulates is an integral of its rate.
The two meanings are the same fact: the total accumulated from a rate is the area under the rate's graph. A car at a steady 20 m/s for 5 seconds travels 100 m, which is the area of a 20 by 5 rectangle; at a changing speed the distance is still the area, now under a curve.
Indefinite and definite integrals
| Indefinite integral | Definite integral | |
|---|---|---|
| Written | ∫ f(x) dx | ∫ₐᵇ f(x) dx |
| Gives | A family of functions F(x) + C whose derivative is f(x) (the antiderivative) | A number: the area or the total from a to b |
| Example | ∫ 2x dx = x² + C | ∫₀³ 2x dx = [x²]₀³ = 9 − 0 = 9 |
| The C | Required: differentiation loses constants, so the reverse cannot recover them | Cancels in the subtraction, so omitted |
| Used for | Finding the function from its rate; solving differential equations; the tool | Areas, volumes, distances, totals, averages; the answer |
The fundamental theorem of calculus
The theorem that makes integrals computable rather than estimated with rectangles: if F is any antiderivative of f (that is, F′ = f), then ∫ₐᵇ f(x) dx = F(b) − F(a). Finding an area is reduced to finding a function whose derivative is the one given, then evaluating it at two points. Its other half says that the function A(x) = ∫ₐˣ f(t) dt, the area accumulated up to x, has derivative f(x): integration and differentiation undo each other. Every rule below is a derivative rule read backwards.
The rules a first course uses
| Rule | Formula | Comes from |
|---|---|---|
| Power rule | ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1 | d/dx of xⁿ⁺¹/(n + 1) is xⁿ |
| Constant | ∫ k dx = kx + C | |
| Constant multiple and sum | ∫ [a f(x) + b g(x)] dx = a ∫ f dx + b ∫ g dx | Integrate term by term |
| The exception to the power rule | ∫ 1/x dx = ln | x |
| Exponential | ∫ eˣ dx = eˣ + C; ∫ eᵏˣ dx = eᵏˣ/k + C | |
| Sine and cosine | ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C | The signs are the derivative rules reversed |
| Other trigonometric | ∫ sec² x dx = tan x + C; ∫ sec x tan x dx = sec x + C | |
| Linear inside | ∫ f(ax + b) dx = F(ax + b)/a + C | The chain rule reversed for a linear inside |
Substitution undoes the chain rule: choose u as the inside function, replace dx with du/u′, integrate in u, substitute back. ∫ 2x(x² + 1)⁵ dx: let u = x² + 1, du = 2x dx, so the integral is ∫ u⁵ du = u⁶/6 + C = (x² + 1)⁶/6 + C.
Integration by parts undoes the product rule: ∫ u dv = uv − ∫ v du. ∫ x eˣ dx: let u = x, dv = eˣ dx, so du = dx, v = eˣ, giving x eˣ − ∫ eˣ dx = x eˣ − eˣ + C. Choose u as the part that gets simpler when differentiated (the LIATE order: logs, inverse trig, algebraic, trig, exponential).
Four worked examples
1. A polynomial, indefinite. ∫ (3x² − 4x + 5) dx = x³ − 2x² + 5x + C. Check by differentiating: 3x² − 4x + 5.
2. An area, definite. The area under y = x² from 0 to 2: ∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3 ≈ 2.67 square units.
3. A total from a rate. Water flows into a tank at r(t) = 6t litres per minute. How much enters in the first 4 minutes? ∫₀⁴ 6t dt = [3t²]₀⁴ = 48 litres. (Not 6 × 4 = 24: the rate is changing, so the total is the area under the line, a triangle of base 4 and height 24.)
4. The area between two curves. Between y = x and y = x² from 0 to 1: the line is above the parabola there, so the area is ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
Where integrals sit in each course
| Course | Where | What is tested |
|---|---|---|
| AP Calculus AB | Units 6 to 8, the second half of the year | Riemann sums, the fundamental theorem, substitution, area between curves, volumes of revolution, accumulation functions; about 40 percent of the exam |
| AP Calculus BC | Units 6 to 8 plus unit 9 to 10 techniques | AB content plus parts, partial fractions, improper integrals, arc length, parametric and polar areas |
| A-level Mathematics | Year 12 (power rule, area), Year 13 (techniques) | Substitution, parts, partial fractions, trigonometric integrals, areas, differential equations |
| IB Analysis and Approaches | SL: basics and area; HL: full techniques | As A-level, with volumes of revolution and kinematics |
| CBSE Class 12 | Chapters 7 and 8 | Indefinite integrals by substitution, parts and partial fractions; definite integrals and their properties; areas; about a quarter of the paper |
| University Calculus 1 and 2 | Calculus 1: the fundamental theorem and substitution; Calculus 2: everything else |
The six mistakes that lose marks
- Dropping + C on an indefinite integral. One mark every time, on every board.
- The power rule on 1/x. ∫ x⁻¹ dx is ln|x|, not x⁰/0.
- Forgetting to divide by the inside's derivative. ∫ e³ˣ dx = e³ˣ/3 + C, not e³ˣ + C.
- Signs in sine and cosine. The integral of sin x is −cos x; the integral of cos x is sin x.
- Not changing the limits when substituting in a definite integral (or, equally, changing them and then also substituting back).
- Area below the axis. A definite integral of a curve that dips below the axis subtracts that part; for the geometric area, split at the crossing and add absolute values.
What integrals are, in the context of the whole first course, is in what is basic calculus; the order to learn the course in is in how to learn Calculus 1.
Download the integration rules sheet (PDF)
Where to get help
A 1-on-1 online calculus class with a tutor who holds a master's in mathematics: integrals taught as areas first and rules second, every line of working read on the shared whiteboard, and the exam board's own questions (AP free-response, A-level, IB, CBSE) from the first week of the topic. The recording and notes after every class, the same tutor through the exam, US$10 to $13 an hour. The free 30-minute first class is a real lesson on the integral the student is stuck on.
Questions parents ask
1What is an integral in calculus?
2What is the difference between a definite and an indefinite integral?
3What are the basic rules of integration?
4Why do you add C?
5Where do integrals come in the school courses?
See how we teach this, 1-on-1 online →
Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.