Home / Calculus / Integrals

Integrals

The art of accumulation — find areas, volumes, and totals by summing infinitely many infinitesimal pieces.

Antiderivatives

An antiderivative (or indefinite integral) of f(x) is any function F(x) whose derivative is f(x):

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

The "+C" is crucial — there are infinitely many antiderivatives, differing by a constant. Some key antiderivatives:

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C   (n ≠ −1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C

More antiderivatives can be found on the formula sheet.

The Definite Integral

The definite integral computes the signed area between f(x) and the x-axis from a to b:

∫ₐᵇ f(x) dx = lim (n→∞) Σ f(xᵢ)Δx

This is a limit of Riemann sums — rectangles approximating the area. The connection to probability is direct: the probability of a continuous random variable falling in [a,b] is exactly ∫ₐᵇ f(x) dx where f is the probability density function.

The Fundamental Theorem of Calculus

The FTC links differentiation and integration — two seemingly opposite operations are inverses:

Part 1: d/dx [∫ₐˣ f(t) dt] = f(x)

Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a)   where F'(x) = f(x)

Example: Evaluate ∫₁³ (2x + 1) dx

Antiderivative: F(x) = x² + x

F(3) − F(1) = (9 + 3) − (1 + 1) = 12 − 2 = 10

Integration Techniques

1. u-Substitution

The integral version of the chain rule. Let u = g(x), du = g'(x)dx:

Example: ∫ 2x·cos(x²) dx

Let u = x², du = 2x dx

∫ cos(u) du = sin(u) + C = sin(x²) + C

2. Integration by Parts

The integral version of the product rule:

∫ u dv = uv − ∫ v du

Example: ∫ x·eˣ dx

u = x, dv = eˣ dx → du = dx, v = eˣ

= xeˣ − ∫ eˣ dx = xeˣ − eˣ + C

3. Partial Fractions

Decompose a rational function into simpler fractions. Requires factoring the denominator first:

∫ dx/(x² − 1) = ∫ [1/(2(x−1)) − 1/(2(x+1))] dx = ½ ln|x−1| − ½ ln|x+1| + C

4. Trigonometric Substitution

Use trig identities to handle expressions involving √(a² − x²), √(a² + x²), or √(x² − a²).

Applications of Integration

Area Between Curves

Area = ∫ₐᵇ [f(x) − g(x)] dx   where f(x) ≥ g(x) on [a,b]

Volume of Revolution

Disk method: V = π ∫ₐᵇ [f(x)]² dx
Shell method: V = 2π ∫ₐᵇ x·f(x) dx

These connect integration to 3D geometric shapes — computing volumes that geometry formulas alone can't handle.

Arc Length

L = ∫ₐᵇ √(1 + [f'(x)]²) dx

Probability

For a continuous probability distribution with density f(x):

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx   and   ∫₋∞^∞ f(x) dx = 1

The normal distribution, exponential distribution, and every other continuous distribution is defined through integrals.

Integration reaches into every field: physics (work, energy, fluid pressure), engineering (signal processing via Fourier transforms), economics (consumer/producer surplus), and linear algebra (inner products on function spaces). The concept of area under a curve is one of the most widely applied ideas in all of science.

Related Topics

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
Derivatives Differentiation is the inverse of integration — know both sides. First-Order DEs Solving DEs often requires integration techniques directly. Polynomials & Factoring Partial fraction decomposition requires factoring skills. Probability Distributions Every continuous probability is computed via integration.
RDVm1U+slcDAQN/rzygdx1yBnBWBd1sNh9Nq3FzP8lt2rWf0gkPrFPtGG2O72R37riQf6GdGIHQTaXFuxI6pNAjFKZKZtJbzuDD/fNx4VIbPOv4AzK9oJFTrsZltEB/qi+q5bnwk59Q0Yx+FqnOEEKuWcDJ3LmivbfPSPqh7Qv/tKXhszDXZuGCH/I2X36Dm2Apna3vf7vzIgg+EaGhE1ZjlSMG5VmlbhI8PzJ8LqUsLZNfz7r+6AocezY8wsbqAKRfLOmqdiILF2loSlZKpYbZ4BQZYngfeYzTI2vXcjgV0lgDbY66o0LUtyw7jND02N4Ql3HsmwTEqg+L0wokR8iU7D/s4OPdtdT8+cotH6LGDiYdOVCf2K5gpBIZ6hb+AmOVx6kRCYkdzhQB4pdXlLmuGFFJV99oxGsR7wiEqcGzN3dQfOwyKEtIQ5CNuiXkD49f+oMYwVXJi9iMCUGdkNISFZFGa9MUw7ZlRwnzRUKYV+/KABCJJBxIpYuK2w8ZGe81+o15o2rDJi5QfUKoNseQKU5HbS6VIV7E5alZHsSSXLDf1QbRyR99g4tqTPrmItWLfvISPE0CFX6QJAsNHJjKgIo0fHgMWsi8Q/5GG+YU+dG9sSi5TpyF9MNmgSWrGfV1Up8W+UHcVQg2ADco+lCakHmC/fKpAXBFs3O2AH8RXePKiiP9MSiAUCcvXp0A69nhiVkAge61xm4ldAIv6oDLEgCSBOLqtleDYUrBxYVkqWOB6LuV9XwfZTdeqpCER2QT9OeBbZ6bJgAWROd+8/kZeq2vnnacxd7WZIa9Yhea6rf1ERpdLjppRjs8rY/QpaJ7prO4Rfa/+7qu4AQWErwggTZ0wFPjTM8IiCTkPFQQP15Uk8rFBDwPf6NV0ePPIaFAP0QmiEsCG0QMAnsiACdEb