How to solve Integrals
Integrals are the sum of infinite summands, infinitely small.
Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral
is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below the x axis subtract from the total.
The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
INTEGRATION BY SUBSTITUTION
Whenever an integral can be written as:
if we change t=u(x), the integral transforms in:
may be easier to solve.
INTEGRATION BY PARTS
This method is useful in the cases where the integrating can put as the product of a function for the differential of other one
INTEGRATION OF RATIONAL FUNCTIONS
A rational function is any function which can be written as the ratio of two polynomial functions.
Proper: if the degree of the polynomial divisor is greater than the dividend.
Improper: if the dividend polynomial degree is greater than or equal to the divisor.
Any improper rational function can be decomposed into the sum of a polynomial plus a proper rational function.
Therefore, the integral of an improper rational function can be written:
To solve the integral of a rational function is decomposed into a sum of simple fractions:
1) The denominator is decomposed into a product of factors as follows:
2) Is then written
and then obtain the following expression:
3) The coefficients A, B, …, N, are determined by successively x = a, x = b, etc.
4) Coefficients obtained, we integrate expression.
Case where the denominator polynomial has multiple roots
INTEGRATION BY TRIGONOMETRIC SUBSTITUTION
It is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions.
You can download the App BioProfe READER to practice this theory with self-corrected exercises.