§ 01What Is Integration?

Integration asks two questions that turn out to have the same answer: what is the area under a curve, and what function has this derivative? The extraordinary fact that these questions are equivalent is the Fundamental Theorem of Calculus.

Start with a concrete problem. You are watching a car move, and you know its velocity v(t) at every instant. How far has it travelled between time a and time b? The answer is the area under the velocity curve, because distance = velocity × time, and when velocity varies you must add up infinitely many infinitely thin rectangles. That sum is an integral.

The Definite Integral — Area as a Limit

∫_a^b f(x) dx = lim_n→∞ ∑_i=1ⁿ f(x_i) Δx

Divide [a, b] into n strips of width Δx = (b−a)/n, take the height f(x_i) of each strip, sum all the areas, and take n → ∞. That is a definite integral.

The indefinite integral is the reverse question: given a function f(x), find a function F(x) whose derivative is f(x). F is called an antiderivative of f. Because any constant differentiates to zero, if F is one antiderivative then F + C (for any constant C) is another, which is why every indefinite integral carries the "+C".

The Indefinite Integral — Antidifferentiation

∫ f(x) dx = F(x) + C where F′(x) = f(x)

The "+C" is not a formality: it represents an entire family of functions, each a valid antiderivative.

Definite vs. Indefinite: The Key Distinction

Definite Integral ∫_a^b f(x) dx

Has limits of integration a and b. The result is a number: the signed area between the curve and the x-axis from a to b.

Output: a real number
No +C needed
Area above x-axis counts positive
Area below x-axis counts negative

Indefinite Integral ∫ f(x) dx

Has no limits. The result is a function (or family of functions), the general antiderivative of f.

Output: a function F(x) + C
Always include +C
Represents all antiderivatives at once
Specific C determined by initial conditions

The Fundamental Theorem of Calculus — The Bridge The Fundamental Theorem says that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) − F(a). This turns the hard problem of computing a limit of sums into the simple problem of evaluating an antiderivative at two points. It is why we bother learning antidifferentiation at all.

§ 02All 12 Integration Topics

Topics are ordered from foundational to advanced. The first three pages build conceptual understanding; pages 4–8 are the core techniques you will use in every course; pages 9–12 are the specialist tools for harder problems.

01 · Foundation Introduction to Integration What an integral means geometrically and algebraically. Antiderivatives, the +C convention, basic antiderivative rules, and the connection to differentiation. 10 examples · quiz 02 · Foundation Riemann Sums Build the definite integral from rectangles. Left, right, and midpoint sums, sigma notation, upper and lower bounds, and the limit process that produces the integral. 10 examples · quiz 03 · Foundation Fundamental Theorem of Calculus Parts 1 and 2. Why differentiation and integration are inverses. d/dx[∫_a^x f(t)dt] = f(x), and how to evaluate definite integrals via F(b)−F(a). 12 examples · quiz 04 · Core Technique The Substitution Rule The reverse chain rule. Let u = g(x), rewrite the integral in u, integrate, substitute back. Covers indefinite and definite integrals, including changing limits. 15 examples · quiz 05 · Core Technique Integration by Parts ∫ u dv = uv − ∫ v du — the reverse product rule. LIATE rule for choosing u, tabular method for rapid repeated IBP, and circular IBP for ∫e^xsin x dx. 15 examples · quiz 06 · Core Technique Partial Fractions Decompose rational functions into simpler fractions before integrating. Covers distinct linear, repeated linear, and irreducible quadratic factors. 12 examples · quiz 07 · Core Technique Trigonometric Integrals Strategies for ∫sin^mx cosⁿx dx, powers of tan and sec, and products of sin/cos with different arguments using product-to-sum identities. 14 examples · quiz 08 · Core Technique Trigonometric Substitution Eliminate square roots like √(a²−x²) using x = a sinθ, √(a²+x²) using x = a tanθ, √(x²−a²) using x = a secθ. Includes completing the square. 12 examples · quiz 09 · Special Cases Improper Integrals Integrate over infinite intervals or past discontinuities by replacing the problematic limit with a variable and taking a limit. Convergence vs. divergence, comparison test. 12 examples · quiz 10 · Special Cases Numerical Integration Trapezoid Rule and Simpson’s Rule — when an exact antiderivative doesn’t exist or is impractical. Error bound formulas, full numerical tables, and when each method excels. 8 examples · quiz 11 · Applications Volumes of Revolution Rotate a curve around an axis and find the volume using the Disk Method V = π∫[f(x)]² dx, the Washer Method, and the Shell Method V = 2π∫x f(x) dx. 12 examples · quiz 12 · Applications Arc Length & Surface Area Measure the length of a curve L = ∫√(1+[f′(x)]²) dx. Arc length in parametric form, and the surface area of revolution formula. 8 examples · quiz

∫ ✦ dx

§ 03Suggested Learning Path

Integration techniques form a deliberate hierarchy. Each stage below assumes the previous one. A common mistake is jumping straight to substitution before the Fundamental Theorem is solid, that shortcut costs hours of confusion later. Follow the stages in order the first time through.

Stage 1

Prerequisites: Differentiation (all rules) · Limits Introduction to Integration Riemann Sums Fundamental Theorem of Calculus

Stage 2

Prerequisites: Stage 1 Substitution Rule Integration by Parts

Stage 3

Prerequisites: Stages 1–2 · Trig identities Partial Fractions Trig Integrals Trig Substitution

Stage 4

Prerequisites: Stages 1–3 · Limits at infinity Improper Integrals Numerical Integration

Stage 5

Prerequisites: All prior stages · Geometric insight Volumes of Revolution Arc Length & Surface Area

§ 04Quick Reference — Standard Integrals & Rules

Every formula below is derived in full on its dedicated topic page. Use this as a revision card once you have worked through the section. If a formula feels arbitrary, go back to the page where it was proved, no formula here was handed down from the sky.

Basic Antiderivative Rules

Power Rule (Reverse)

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C Valid for n ≠ −1. "Increase the power by 1, divide by the new power."

Constant Multiple

∫ c⋅f(x) dx = c ∫ f(x) dx Constants factor out of integrals, just as in differentiation.

Sum / Difference

∫ [f ± g] dx = ∫ f dx ± ∫ g dx Integrate term by term — works for finite sums only.

Natural Log

∫ (1/x) dx = ln|x| + C The case n = −1 excluded by the power rule. The |x| handles negative x.

Exponential

∫ e^x dx = e^x + C eˣ is its own antiderivative — the same magical fact as its derivative.

General Exponential

∫ a^x dx = a^x/(ln a) + C The ln a denominator is the reciprocal of the factor in its derivative.

Trigonometric Integrals

sin x & cos x

∫ sin x dx = −cos x + C ∫ cos x dx = sin x + C Note the minus sign on the sin integral — the most common slip.

tan x & cot x

∫ tan x dx = ln|sec x| + C ∫ cot x dx = ln|sin x| + C Both derived by substitution: let u = cos x or u = sin x.

sec x & csc x

∫ sec x dx = ln|sec x + tan x| + C ∫ csc x dx = −ln|csc x + cot x| + C These non-obvious forms are derived by multiplying by a clever fraction.

sec² x & csc² x

∫ sec² x dx = tan x + C ∫ csc² x dx = −cot x + C Immediate from the derivatives of tan and cot.

Inverse Trig Integrals

arcsin form

∫ 1/√(a²−x²) dx = arcsin(x/a) + C Valid for |x| < a. Arises frequently in trig substitution problems.

arctan form

∫ 1/(a²+x²) dx = (1/a) arctan(x/a) + C The (1/a) factor is easily forgotten. Verify by differentiating the result.

arcsec form

∫ 1/(x√(x²−a²)) dx = (1/a) arcsec(|x|/a) + C Less common but appears in trig substitution with √(x²−a²).

Key Techniques — When to Use Each

Substitution Rule

∫ f(g(x))⋅g′(x) dx = ∫ f(u) du Use when you see a composite function and its derivative multiplied together. Set u = inner function.

Integration by Parts

∫ u dv = uv − ∫ v du Use for products of unlike functions: polynomial × trig, polynomial × e^x, or e^x × trig.

Partial Fractions

P(x)/Q(x) = A/(x−r) + B/(x−s) + … Use for rational functions where deg(P) < deg(Q) and Q factors into linear or quadratic terms.

Trig Substitution

√(a²−x²): x = a sinθ Use when the integrand contains a square root of a sum or difference involving x². Complete the square first if needed.

§ 05How to Choose the Right Technique

The hardest part of integration is not executing the technique, it is recognising which technique to use. The flowchart below is the mental algorithm experienced mathematicians run through (quickly and almost subconsciously) when facing a new integral.

Integration Decision Guide

Is it a standard form? → Apply directly
Can you simplify algebraically first? → Expand, factor, or divide
Do you see f(g(x))⋅g′(x)? → Substitution
Is it a product of unlike functions? → Integration by Parts
Is it a rational function P/Q? → Partial Fractions (after factoring Q)
Does it contain √(a²±x²) or √(x²−a²)? → Trig Substitution
Is the interval infinite or the integrand unbounded? → Improper Integral

In practice, many integrals require combining techniques: for example, a substitution that produces a rational function requiring partial fractions, or an integration by parts that leaves a trig integral. The more worked examples you study, the faster pattern recognition becomes. There is no shortcut for the examples themselves.

The Most Important Habit: Always Verify by Differentiating After computing ∫ f(x) dx = F(x) + C, differentiate F(x) and check that you get f(x) back. This takes thirty seconds and catches the vast majority of errors — wrong signs, missing chain-rule factors, and dropped constants. Make it automatic.

§ 06Where Integration Leads

Mastering single-variable integration is the gateway to the rest of advanced mathematics. Every section below requires fluent integration as a prerequisite, these are not optional extensions, they are the next chapters in the story.

Solve with It

Differential Equations

Power Series

Taylor & Maclaurin Series

Extend It

Double & Triple Integrals

Work & Flux

Line & Surface Integrals

Applications

Continuous Probability

Integration and Differential Equations — An Inseparable Pair The simplest differential equation dy/dx = f(x) is solved by integrating both sides. Every separable equation, every first-order linear ODE, every Laplace transform uses integration at its core. Once you can integrate fluently, the entire differential equations section opens up — and with it, the language used to model virtually everything in physics, biology, economics, and engineering.