Indefinite integral . It is the set of infinite primitives that a function can have .
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- 1 Definition
- 1 Indefinite integral
- 2 immediate integrals
- 3 Properties
- 1 Examples
- 4 Integration methods
- 5 Applications
- 6 See also
- 7 Sources
Integrating is the reciprocal process of deriving , that is, given a function f (x), it looks for those functions F (x) that when derived lead to f (x).
F (x) is then said to be a primitive or antiderivative of f (x); In other words, the primitives of f (x) are the derivable functions F (x) such that:
F ‘(x) = f (x).
If a function f (x) has a primitive, it has infinite primitives, all of which differ in a constant.
[F (x) + C] ‘= F’ (x) + 0 = F ‘(x) = f (x)
Indefinite integral is the set of infinite primitives that a function can have.
It is represented by ∫ f (x) dx .
It is read as “the indefinite integral of f (x) with respect to x” Therefore, f (x) dx is a set of functions; it is not a single function, nor a number.
The function f being integrated is called the integrand, and the variable x is called the integration variable.
C is the integration constant and can take any real numerical value.
If F (x) is a primitive of f (x) we have to:
∫ f (x) dx = F (x) + C
To verify that the primitive of a function is correct, simply derive.
If u = x (u ‘= 1), we have a table of simple integrals:
- X cos x dx = sin x + C
- The integral of the product of a constant times a function is equal to the constant times the integral of the function.
∫ kf (x) dx = k ∫f (x) dx
- The integral of a sum of functions is equal to the sum of the integrals of those functions.
∫ [f (x) + g (x)] dx = ∫ f (x) dx + ∫ g (x) dx
Integration methods are understood as any of the different elementary techniques used to calculate an antiderivative or indefinite integral of a function.
Thus, given a function f (x), integration methods are techniques whose use (usually combined) allows finding a function F (x) such that
F (x) = ∫ ƒ (x) dx
which, by the fundamental theorem of the calculation is equivalent to finding a function F (x) such that f (x) is its derivative .
There are several methods among which the following stand out:
- Integration by Change of variable.
- Integration by partto integrate function products.
- trigonometric functions.
- Integration of rational functions.
With an integral you can calculate magnitudes as diverse as areas , volumes , curve lengths , the work done by a force, the mass of a solid , moments of inertia , the electric field , the flow of a fluid through a surface and many plus. It is notable, however, that the way of proceeding is almost always the same, and consists in expressing the exact value of the quantity to be calculated as a limit of Riemann sums , to deduce, from them, the integral whose calculation provides the solution to the problem.