## How to solve Integrals

# 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

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:

**KNOWN INTEGRALS**

**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.

**THEOREM**

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.

For example:

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.

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