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

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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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.

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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 |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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KNOWN INTEGRALS

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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INTEGRATION BY SUBSTITUTION

 

Whenever an integral can be written as:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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if we change t=u(x), the integral transforms in:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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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 |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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INTEGRATION OF RATIONAL FUNCTIONS

 

A rational function is any function which can be written as the ratio of two polynomial functions.

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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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.

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Therefore, the integral of an improper rational function can be written:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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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 |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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2) Is then written

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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and then obtain the following expression:

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

 

For example:

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4) Coefficients obtained, we integrate expression.

 

Case where the denominator polynomial has multiple roots

Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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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 |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | How to solve an integral | Integrals

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You can download the App BioProfe READER to practice this theory with self-corrected exercises.

 

 

 

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