Integration
Integration is the process of finding the integral of a function — the reverse operation of differentiation. It computes the accumulation of quantities over an interval and is used to find areas, volumes, total distances, probabilities, and many other quantities in science and engineering. Integration encompasses a wide range of techniques including substitution, integration by parts, partial fractions, and numerical methods.
Integration Techniques and Their Applications
| Technique | When to Use | Key Formula | Example Integrand | Difficulty |
|---|---|---|---|---|
| Basic Rules | Polynomials, exponentials, trig | ∫xⁿ dx = xⁿ⁺¹/(n+1) | x³ + 2x | Easy |
| Substitution | Composite functions | ∫f(g(x))g'(x) dx | sin(x²)·2x | Medium |
| Integration by Parts | Products of functions | ∫u dv = uv − ∫v du | x·eˣ | Medium |
| Partial Fractions | Rational functions | Decompose into simpler fractions | 1/(x²−1) | Medium–Hard |
| Trigonometric Substitution | Radicals involving a²±x² | x = a·sin(θ) or a·tan(θ) | √(1−x²) | Hard |
| Numerical Integration | Non-elementary integrands | Simpson's rule, Trapezoidal | eˣ² | Varies |
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Related Terms
Integral
An integral is a mathematical object that represents the accumulation of quantities, such as areas under curves, total displacement, or accumulated change. There are two main types: the definite integral, which yields a numerical value representing the net area between a function and the x-axis over an interval, and the indefinite integral, which yields a family of antiderivative functions. Integration is the reverse process of differentiation and is one of the two fundamental operations of calculus.
Definite Integral
A definite integral is an integral evaluated over a specific closed interval [a, b], producing a single numerical value that represents the net signed area between the function's curve and the x-axis over that interval. It is defined as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals are used extensively in physics for calculating work, displacement, charge, and probability.
Integration by Substitution
Integration by substitution is a technique that simplifies complex integrals by replacing a part of the integrand with a new variable, effectively reversing the chain rule of differentiation. The method works by identifying an inner function u = g(x), computing du = g'(x) dx, and rewriting the integral entirely in terms of u. It is one of the most widely used integration techniques and is applicable whenever an integrand contains a composite function paired with its derivative.
From Latin "integrare" meaning "to make whole." The systematic study of integration was formalized in the 17th century by Newton and Leibniz as the inverse of differentiation, forming one pillar of the fundamental theorem of calculus.