Integrals

Antiderivatives

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

∫ 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.

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