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Integration is a mathematical concept that involves finding the area under a curve. It is a fundamental tool in calculus and is used to solve various problems, such as finding the distance traveled by an object or the volume of a 3D shape.
Differentiation and integration are inverse operations. While differentiation finds the slope of a curve at a specific point, integration finds the area under the curve. In other words, differentiation is about finding the rate of change, while integration is about finding the total change.
There are two main types of integration: indefinite and definite. Indefinite integration involves finding the general antiderivative of a function, while definite integration involves evaluating the area under a curve between two specific points. Other types of integration include improper integration, multiple integration, and line integration.
The basic integration rules include the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant multiple rule, which states that the integral of c*f(x) is c times the integral of f(x). Other important rules include the sum and difference rules, the substitution rule, and the integration by parts rule.
Integration is used in various fields, including physics, engineering, and economics. It is used to calculate areas, volumes, and rates of change, making it an essential tool for solving real-world problems. For example, integration is used to calculate the trajectory of a projectile, the volume of a tank, and the profit of a business.