integration of functions

integration of functions

1. ¿What is it ?

   • Integration is a fundamental concept of calculation and mathematical analysis. Basically, an integral is a generalization of the sum of infinite, infinitesimally small addends: a continuous sum. The integral is the inverse operation to the derivative
     • discovery
       • in the centuryxvll by isaac newton and gottried leiniz
         • ¿what is it for?
           • It is mainly used for the calculation of volume areas of regions and solids of revolution.

2. ¿what does it consist of ?

   • integration is the process of finding the integral of a function
     • basis
       • the calculation theorem indicates that the derivation and integration are inverse operations F '(x) = F (x)
         • ¿how to integrate a function ?
           • through different methods that allow us to find the integral or antiderivative of a given function

3. types of integration

   • direct integration
     • integration by parts
       • integration by substitution
         • trigonometric substitution
           • integration by partial fractions
             • Partial fractions is an integration method that allows us to solve integrals of certain rational functions that cannot be solved by the other methods.
             • definite integral
               • is
                 • It is a concept used to determine the value of the areas bounded by curves and lines. Given the interval [a, b] in which, for each of its points x, a function f (x) is defined that is greater than or equal to 0 in [a, b], it is called the definite integral of the function between the points a and bal area of ​​the portion of the plane that is limited by the function, the horizontal axis OX and the vertical lines of equations x = a and x = b
                   • Properties of the definite integral
                     • Every integral extended to a single point interval, [a, a], is equal to zero.
                     • When the function f (x) is greater than zero, its integral is positive; if the function is less than zero, its integral is negative.
                     • The integral of a sum of functions is equal to the sum of its integrals taken separately.
                     • By exchanging the limits of an integral, it changes sign.
                     • Given three points such that a <b <c, then it is true that (piecewise integration):
                   • example
           • Trigonometric substitution allows transforming one integral into another that contains trigonometric functions whose integration process is easier.
         • The method of integration by substitution or change of variable is based on the chain rule. The method is based on identifying a part of what is going to be integrated with a new variable t, so that a simpler integral is obtained.
       • Integration by part. The integration method by parts allows to calculate the integral of a product of two functions applying the formula: ∫ f (x) g '(x) dx = f (x) g (x) - ∫ f' (x) g (x) dx
     • They are the integrals that do not require applying any integration method because they are very simple. For example, the integral of 2x is x2 + C, where C is the integration constant.

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