# Integrals theory

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 Bioprofe |To solve an integral | 01

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. Bioprofe |To solve an integral | 02

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: Bioprofe |To solve an integral | 03

KNOWN INTEGRALS Bioprofe |To solve an integral | 04 Bioprofe |To solve an integral | 05 Bioprofe |To solve an integral | 06 Bioprofe |To solve an integral | 07 Bioprofe |To solve an integral | 08 Bioprofe |To solve an integral | 09 Bioprofe |To solve an integral | 10 Bioprofe |To solve an integral | 11 Bioprofe |To solve an integral | 12 Bioprofe |To solve an integral | 13 Bioprofe |To solve an integral | 14 Bioprofe |To solve an integral | 15 Bioprofe |To solve an integral | 16 Bioprofe |To solve an integral | 17 Bioprofe |To solve an integral | 18 Bioprofe |To solve an integral | 19 Bioprofe |To solve an integral | 20

INTEGRATION BY SUBSTITUTION

Whenever an integral can be written as: Bioprofe |To solve an integral | 21

if we change t=u(x), the integral transforms in: Bioprofe |To solve an integral | 22

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 Bioprofe |To solve an integral | 23

INTEGRATION OF RATIONAL FUNCTIONS

A rational function is any function which can be written as the ratio of two polynomial functions. Bioprofe |To solve an integral | 24

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.

THEOREM

Any improper rational function can be decomposed into the sum of a polynomial plus a proper rational function. Bioprofe |To solve an integral | 25

Therefore, the integral of an improper rational function can be written: Bioprofe |To solve an integral | 26

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: Bioprofe |To solve an integral | 27

2) Is then written Bioprofe |To solve an integral | 28

and then obtain the following expression: Bioprofe |To solve an integral | 29

3) The coefficients A, B, …, N, are determined by successively x = a, x = b, etc.

For example: Bioprofe |To solve an integral | 30

4) Coefficients obtained, we integrate expression.

Case where the denominator polynomial has multiple roots Bioprofe |To solve an integral | 31 Bioprofe |To solve an integral | 32

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. Bioprofe |To solve an integral | 33 Bioprofe |To solve an integral | 34 Bioprofe |To solve an integral | 35 Bioprofe |To solve an integral | 36 Bioprofe |To solve an integral | 37 Bioprofe |To solve an integral | 38

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