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Integrals



The Idea of Accumulation
Two Types of Integrals
Notation and Terminology
Definite Integrals
Indefinite Integrals
Integration Rules
Integration Techniques
Special Integrals
Improper Integrals
Evaluating Integrals
Summary: A Map of the Integrals Subtree



The Art of Accumulation


Differentiation asks: given a quantity, how fast is it changing? Integration asks the reverse: given a rate of change, what is the total accumulated quantity? These two questions form the twin pillars of calculus.

The integral sign ∫\int represents summation taken to its limit—adding infinitely many infinitely small pieces. When you integrate a velocity function, you recover distance traveled. When you integrate a density function, you recover total mass. The pattern repeats across physics, economics, and probability: rates become totals through integration.

Two distinct but related concepts share this notation. The definite integral

∫abf(x) dx\int_a^b f(x)\, dx


computes a number—the signed area under a curve, the accumulated quantity between two bounds. The indefinite integral

∫f(x) dx\int f(x)\, dx


finds a function—the antiderivative whose derivative returns the original integrand. The Fundamental Theorem of Calculus reveals these as two faces of the same idea.

Key Terms

Indefinite

Antiderivative— a function FF with F′(x)=f(x)F'(x) = f(x)
Indefinite Integral— the family F(x)+CF(x) + C of all antiderivatives

Definite

Definite Integral— accumulated signed area over [a,b][a, b]
Riemann Sum— rectangular approximation whose limit defines the integral
Bounds of Integration— lower and upper limits aa, bb
Signed Area— area above the axis minus area below
Average Value of a Function— 1b−a∫abf(x) dx\frac{1}{b-a}\int_a^b f(x)\,dx

Components

Integrand— the function being integrated

Extensions

Improper Integral— infinite interval or unbounded integrand

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Two Types of Integrals


The integral symbol carries two distinct meanings depending on context.

Definite Integrals


∫abf(x) dx\int_a^b f(x)\, dx


The limits aa and bb bound the region of integration. The result is a number representing accumulated quantity—area, volume, total change, or another aggregate measure.

Indefinite Integrals


∫f(x) dx=F(x)+C\int f(x)\, dx = F(x) + C


No limits appear. The result is a family of functions—all antiderivatives of f(x)f(x), differing by an arbitrary constant CC.

The same symbol serves both purposes. Context determines which interpretation applies: limits present means definite, limits absent means indefinite.
Aspect Definite integral ∫ab f(x) dx Indefinite integral ∫ f(x) dx
Bounds of integration present (a and b) absent
Result type a number a family of functions F(x) + C
Geometric meaning signed area between f and the x-axis on [a, b] no fixed geometry — represents all antiderivatives
Constant of integration cancels in F(b) − F(a) essential — the "+ C"
Typical use accumulated quantity (distance, work, mass, probability) finding the function whose derivative is f
Connection evaluated via FTC: F(b) − F(a) supplies the F used by FTC

Notation and Terminology


The integral notation carries specific meaning in each component.

The integral sign ∫\int is an elongated S, standing for "sum." It originated with Leibniz, who conceived integration as summing infinitesimal pieces.

The integrand f(x)f(x) is the function being integrated. It describes what is accumulated at each point.

The differential dxdx indicates the variable of integration and represents an infinitesimal width. Together, f(x) dxf(x)\, dx represents an infinitesimal contribution to the total.

For definite integrals, the limits of integration aa and bb specify where accumulation begins and ends:

∫abf(x) dx\int_a^b f(x)\, dx


The lower limit aa appears at the bottom of the integral sign, the upper limit bb at the top.

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Definite Integrals


The definite integral computes signed area—the area between a curve and the xx-axis, with regions below the axis counted as negative.

∫abf(x) dx\int_a^b f(x)\, dx


Positive where f(x)>0f(x) > 0, negative where f(x)<0f(x) < 0. The integral sums these signed contributions.

Rigorously, the definite integral arises as a limit of Riemann sums. Partition the interval [a,b][a, b] into subintervals, approximate the area with rectangles, and take the limit as the rectangles become infinitely thin. This construction gives precise meaning to "area under a curve."
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Indefinite Integrals


The indefinite integral reverses differentiation. If F′(x)=f(x)F'(x) = f(x), then FF is an antiderivative of ff, and we write:

∫f(x) dx=F(x)+C\int f(x)\, dx = F(x) + C


The constant CC is essential. Since the derivative of any constant is zero, infinitely many functions share the same derivative. The "+C+ C" represents this entire family.

Finding antiderivatives is the core skill of indefinite integration. Unlike differentiation, which follows systematic rules, integration often requires insight, technique, and pattern recognition.
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Integration Rules


Integration rules provide the algebraic structure for computing integrals.

Linearity allows integrals to be split and scaled:

∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx\int [f(x) + g(x)]\, dx = \int f(x)\, dx + \int g(x)\, dx


∫c⋅f(x) dx=c∫f(x) dx\int c \cdot f(x)\, dx = c \int f(x)\, dx


The Fundamental Theorem of Calculus connects definite and indefinite integrals. If FF is any antiderivative of ff:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)


This theorem transforms the problem of computing areas into the problem of finding antiderivatives—a profound simplification.
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Integration Techniques


Not every function yields to direct antidifferentiation. Integration techniques provide methods for transforming difficult integrals into manageable ones.

Substitution reverses the chain rule. Identify an inner function and its derivative, change variables, and integrate.

Integration by parts reverses the product rule:

∫u dv=uv−∫v du\int u\, dv = uv - \int v\, du


Partial fractions decompose rational functions into simpler pieces. Trigonometric substitution handles integrals involving square roots of quadratics.

Choosing the right technique is an acquired skill—pattern recognition developed through practice.
Technique What it reverses Typical setup Best when
Substitution chain rule let u = g(x); du = g'(x) dx an inner function and its derivative both appear in the integrand
Integration by parts product rule ∫ u dv = uv − ∫ v du integrand is a product of unrelated factors (e.g., x · eˣ, x · ln x)
Partial fractions addition of fractions decompose P(x)/Q(x) into simpler rational pieces integrand is a rational function with factorable denominator
Trigonometric substitution Pythagorean identities x = a sin θ, a tan θ, or a sec θ depending on the radical integrand contains √(a² − x²), √(a² + x²), or √(x² − a²)
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Special Integrals


Certain special integrals appear so frequently that memorizing them accelerates all subsequent work.

∫xn dx=xn+1n+1+C(n≠−1)\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)


∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\, dx = \ln|x| + C


∫ex dx=ex+C\int e^x\, dx = e^x + C


∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C\int \sin x\, dx = -\cos x + C \qquad \int \cos x\, dx = \sin x + C


∫11+x2 dx=arctan⁡x+C\int \frac{1}{1 + x^2}\, dx = \arctan x + C


These formulas serve as building blocks. Complex integrals often reduce to combinations of these standard forms.
Integrand Antiderivative Note
xn  (n ≠ −1) xn+1 / (n+1) + C power rule for integrals
1 / x ln|x| + C handles the n = −1 case excluded above
ex ex + C its own antiderivative
sin x −cos x + C sign flips relative to the derivative
cos x sin x + C no sign change
1 / (1 + x²) arctan x + C reverses the derivative of arctan
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Improper Integrals


Standard definite integrals require finite intervals and bounded integrands. Improper integrals extend integration beyond these constraints.

Integrals over infinite intervals:

∫1∞1x2 dx=lim⁡b→∞∫1b1x2 dx=1\int_1^{\infty} \frac{1}{x^2}\, dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2}\, dx = 1


Integrals of unbounded functions:

∫011x dx=lim⁡a→0+∫a11x dx=2\int_0^1 \frac{1}{\sqrt{x}}\, dx = \lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}}\, dx = 2


An improper integral converges if the limit exists and is finite; otherwise it diverges. The distinction matters—some infinite regions have finite area, others do not.
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Evaluating Integrals


Evaluating integrals combines all the preceding ideas: recognize the form, apply appropriate techniques, and verify the result.

Start with direct antidifferentiation. If the integrand matches a known form, apply the formula. If not, consider substitution, parts, or other techniques.

For definite integrals, set up the problem correctly: identify bounds, express the integrand in terms of the integration variable, and apply the Fundamental Theorem.

Verification is simple: differentiate your answer. The derivative of the antiderivative should return the original integrand. This check catches algebraic errors and sign mistakes.
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Summary: A Map of the Integrals Subtree


The seven subtopics covered in the sections above are different facets of the same underlying tool: recovering totals from rates, and reversing differentiation. The table below collects them in one place, pairing each subtopic with its key idea, the situation that calls for it, and a direct link to its dedicated page. Read it as a navigation map for the rest of this chapter — the row matching a given problem points to where to start.
Subtopic Key idea When you need it Subpage
Definite integrals signed area under a curve via limit of Riemann sums computing an accumulated total over an interval definite
Indefinite integrals the family F(x) + C of antiderivatives of f recovering a function from its derivative indefinite
Integration rules linearity and the Fundamental Theorem of Calculus splitting integrals and evaluating with antiderivatives rules
Techniques substitution, parts, partial fractions, trig substitution the integrand does not match a known form directly techniques
Special integrals standard formulas worth memorizing recognizing common building-block patterns special
Improper integrals integrals over infinite intervals or with unbounded integrands interval is unbounded or integrand blows up improper
Evaluating integrals recognize, apply, verify by differentiating the result working a concrete integral from start to finish evaluating