Given a function f(x), if there exists a function F(x) that satisfies F'(x)=f(x), then F(x) is said to be an original function of f(x). We usually use ∫f(x)dx to represent the indefinite integral of f(x), which represents the set of all original functions, that is, ∫f(x)dx is the set of original functions of f(x).
The main application of indefinite integrals is to solve definite integrals of functions. A definite integral is the integral evaluation of a function over a given interval. If we can find the original function of the function, then we can solve the definite integral problem conveniently. In addition, the indefinite integral has a wide range of applications in calculus, physics, engineering, etc.
Polynomial division:
Substitution Integral Method (Collect)
Integral method with substitution of elements of the second kind
The following table continues the previous content.
Integration by parts (important)
Integral of Rational Functions
What is a rational function?
A rational function refers to a function of the form P(x)/Q(x), where both P(x) and Q(x) are polynomial functions, and Q(x) is not zero. Here P(x) and Q(x) can be polynomials with real coefficients or complex coefficients.
Rational functions play an important role in mathematics. They have a wide range of applications in algebra, analysis, cybernetics, and more. Properties of rational functions include domain, zero point (making the numerator zero), extremum point, irreducibility, etc.
It should be noted that in rational functions, domain problems arise when Q(x) is zero. In addition, rational functions may have vertical and horizontal asymptotes, which are also their special properties
