INTEGRATION OF SOME RATIONAL AND IRRATIONAL FUNCTIONS USING SOFTWARE - - PowerPoint PPT Presentation

INTEGRATION OF SOME RATIONAL AND IRRATIONAL FUNCTIONS USING SOFTWARE “MATHCAD”

5-th international scientific conference “APPLIED INFORMATION AND COMMUNICATION TECHNOLOGIES 2012” 26-27 April 2012, Latvia University of Agriculture

Estonia University of Life Sciences Latvia University of Agriculture OLGA PANOVA, SVETLANA ATSLEGA

INTRODUCTION

2

Mathematics for the engineer. Engineering is based on mathematics, physics and technological solutions. Computing technology is often included in these basic science. The PC change radically the nature of the teaching of mathematics. As computers have become more powerful so the problems which are being tackled in the real world are becoming ever more complex, demanding more sophisticated numerical techniques.

INTRODUCTION

3

• Linear algebra
• Matrices and determinants
• Vector algebra and applications
• Analysis and Calculus
• Functions and their inverses
• Differentiation
• Indefinite integration
• Definite integration, applications to areas and volumes
• Ordinary differential equations
• Sequences, series
• Statistics and Probability

INTRODUCTION

4

Indefinite integration Double integrals Ordinary differential equations Partial differential equations Definite integration, applications to areas and volumes

MATHEMATICAL SOFTWARE

5

System Creator Mathcad Parametric Technology Corporqation (1985) Matlab MathWorks (1989) Mathematica Wolfram Research (1986) Maple Symbolic Computation Group, University of Waterloo (1984) Wolfram Alpha Wolfram Research (2009)

INDEFINITE INTEGRATION

6

One of the most important classes of elementary functions, whose integrals can be found in comparatively simple way and always are elementary functions, are rational functions. Except for a few very special cases, currently we have no way to find the integral of a general rational function. Integration of rational functions Integration of irrational functions

INTEGRATION OF RATIONAL FUNCTIONS

Example 1. Calculate

1 𝑦2+𝑞𝑦+𝑟 𝑒𝑦, where 𝑞, 𝑟 ∈ 𝑆.

• Remark. The computer algorithm might simplify expressions different

from what we are used to. Also, it does not give the integration constant “+𝐷”.

7

The polynomial has a double root or is positive

INTEGRATION OF RATIONAL FUNCTIONS

8

Example 2. Calculate the indefinite integral

𝑒𝑦 𝑦2−3𝑦+2.

The polynomial has two different roots. The method of indefinite coefficients The rational function can be written in the form of the partial fractions decomposition

INTEGRATION OF RATIONAL FUNCTIONS

9

Example 4. Calculate the indefinite integral

11𝑦3−107𝑦+108 𝑦4+𝑦3−30𝑦2+76𝑦−56 𝑒𝑦.

Time to get the result The method of partial fractions

INTEGRATION OF IRRATIONAL FUNCTIONS

10

Example 5. Calculate the indefinite integral

1 𝑦+ 𝑦

3

𝑒𝑦

𝑦

𝑜

= 𝑦

1 𝑜

INTEGRATION OF IRRATIONAL FUNCTIONS

11

Example 5. Calculate the indefinite integral

1 𝑦+ 𝑦

3

𝑒𝑦

INTEGRATION OF IRRATIONAL FUNCTIONS

12

INTEGRATION OF IRRATIONAL FUNCTIONS

13

Example 6. Calculate the indefinite integral

1 (𝑦+1) 𝑦2+4𝑦+2 𝑒𝑦

INTEGRATION OF IRRATIONAL FUNCTIONS

14

INTEGRATION OF IRRATIONAL FUNCTIONS

15

Mathcad could not simplify it fully. If student having applied knowledge about the expression h(t) transformation then the Mathcad get the integration result

CONCLUSIONS

16

• Software “Mathcad” is very convenient for computing integrals of

rational or irrational functions.

• Integration of an elementary function not always leads to an elementary
• function. As shown in examples there exist elementary functions whose

integrals are inexpressible in terms of elementary functions without substitution.

• The method of partial fractions is used not only for finding integrals, but

also for analyzing linear differential systems like resonant circuits and feedbackcontrol systems (electrical or mechanical engineering).

• For receiving results in Mathcad in convenient form students are

required theoretical knowledge about integration of some rational and irrational functions.

• The lecturer can suggest to students to use other software: “Matlab”,

“Mathematica”, but these software are more complicated and require more time for obtaining skills for work.

• Software “Mathcad” advantage is the possibility easy and quickly write

the received results.