Integration answers two questions that turn out to be the same: what is the accumulated total of a changing quantity, and what is the area under its graph? The Fundamental Theorem of Calculus welds them to differentiation, and the technique toolkit — substitution, parts, partial fractions — is really a set of inverses to differentiation rules. This page is the standard first-year integration course, organized around that structure.
Partition [a,b], pick sample points, form the Riemann sum \sum f(x_i^*) \Delta x_i, and take the limit as the mesh goes to zero. When that limit exists regardless of the choices, f is integrable and the limit is \int_a^b f(x)\,dx. Every continuous function (and every bounded function with finitely many discontinuities) is integrable. The definition matters beyond pedantry: Riemann sums are how integrals are recognized in disguise (limits of sums in probability, work, and average-value problems) and how numerical quadrature is derived. For the deeper theory where Riemann integration breaks down, see Measure Theory.
Two statements, both essential:
FTC I. If f is continuous on [a,b], then F(x) = \int_a^x f(t)\,dt is differentiable and F'(x) = f(x) — accumulation functions are antiderivatives.
FTC II. If F is any antiderivative of f, then
FTC II is the computational engine; FTC I is the conceptual one (it is why solutions of differential equations can be defined as integrals, and it differentiates integrals with variable limits via the chain rule: \frac{d}{dx}\int_{a}^{g(x)} f = f(g(x))\,g'(x)).
Substitution (inverse chain rule): set u = g(x), du = g'(x)dx. The skill is recognizing the derivative of the inside function sitting in the integrand. For definite integrals, transform the limits and never go back.
Integration by parts (inverse product rule):
Choose u by LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential — pick u from the earliest class present). Classic patterns: \int x e^x dx (one pass), \int x^2 \sin x\,dx (two passes), \int e^x \cos x\,dx (two passes then solve for the integral), \int \ln x\,dx (take dv = dx).
Partial fractions for rational functions: factor the denominator, decompose into terms of the form A/(x-r)^k and (Bx + C)/(x^2 + px + q)^k, integrate each to logs and arctangents. Degree of numerator \ge denominator? Polynomial-divide first.
Trigonometric substitution for \sqrt{a^2 - x^2}, \sqrt{a^2 + x^2}, \sqrt{x^2 - a^2}: substitute x = a\sin\theta, a\tan\theta, a\sec\theta respectively — geometric in origin (parametrizing circles and hyperbolas).
No toolkit is complete: \int e^{-x^2} dx has no elementary antiderivative — which is precisely why the normal distribution's CDF is defined as a special function and evaluated numerically.
Two types, both defined as limits: infinite intervals (\int_1^\infty) and unbounded integrands (\int_0^1 x^{-1/2} dx). The governing benchmark is the p-test: \int_1^\infty x^{-p} dx converges iff p > 1, while \int_0^1 x^{-p} dx converges iff p < 1 — opposite regimes at the two ends, worth internalizing. Comparison tests mirror the series versions, and the integral test makes the series–integral connection exact.
All are Riemann-sum arguments: area between curves (\int (f - g)), volumes by cross-section and by shells, arc length \int \sqrt{1 + (f')^2}\,dx, average value \frac{1}{b-a}\int_a^b f, work and center of mass. In probability, expectations are integrals against densities — the bridge to Random Variables and Distributions.
When symbolic techniques fail (most real integrands), numerical quadrature takes over — trapezoid and Simpson's rules with their O(h^2) and O(h^4) error orders are Taylor-expansion corollaries; see Numerical Methods.