Estonia University of Life Sciences Latvia University of Agriculture O LGA P ANOVA , S VETLANA A TSLEGA 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
I NTRODUCTION 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. 2
I NTRODUCTION • 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 3
I NTRODUCTION Indefinite integration Partial differential Definite integration, equations applications to areas and volumes Ordinary differential Double integrals equations 4
M ATHEMATICAL SOFTWARE 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) 5
I NDEFINITE INTEGRATION 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 6
I NTEGRATION OF RATIONAL FUNCTIONS 1 𝑦 2 +𝑞𝑦+𝑟 𝑒𝑦 , where 𝑞, 𝑟 ∈ 𝑆 . Example 1. Calculate The polynomial has a double root or is positive Remark. The computer algorithm might simplify expressions different from what we are used to. Also, it does not give the integration constant “ +𝐷 ” . 7
I NTEGRATION OF RATIONAL FUNCTIONS 𝑒𝑦 Example 2. Calculate the indefinite integral 𝑦 2 −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 8
I NTEGRATION OF RATIONAL FUNCTIONS 11𝑦 3 −107𝑦+108 𝑦 4 +𝑦 3 −30𝑦 2 +76𝑦−56 𝑒𝑦 . Example 4. Calculate the indefinite integral Time to get the result The method of partial fractions 9
I NTEGRATION OF IRRATIONAL FUNCTIONS 1 𝑒𝑦 Example 5. Calculate the indefinite integral 3 𝑦+ 𝑦 1 = 𝑦 𝑜 𝑜 𝑦 10
I NTEGRATION OF IRRATIONAL FUNCTIONS 1 𝑒𝑦 Example 5. Calculate the indefinite integral 3 𝑦+ 𝑦 11
I NTEGRATION OF IRRATIONAL FUNCTIONS 12
I NTEGRATION OF IRRATIONAL FUNCTIONS 1 (𝑦+1) 𝑦 2 +4𝑦+2 𝑒𝑦 Example 6. Calculate the indefinite integral 13
I NTEGRATION OF IRRATIONAL FUNCTIONS 14
I NTEGRATION OF IRRATIONAL FUNCTIONS Mathcad could not simplify it fully. If student having applied knowledge about the expression h ( t ) transformation then the Mathcad get the integration result 15
C ONCLUSIONS • 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 16 the received results.
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