A mathematical model for integrating product of two functions
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This paper introduces a mathematical model for integrating the product of two functions, which eliminates the need for repetitive applications of the integration by parts method, thereby saving computational time. The model applies when one function is a polynomial of degree n and the other can be integrated at least (n+1) times, ultimately yielding results consistent with traditional methods. Several examples demonstrate the model's effectiveness in providing timely solutions without sacrificing accuracy.
A mathematical model for integrating product of two functions
- 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 A Mathematical Model for Integrating Product of Two Functions M.O. Oke Department of Electrical & Electronic Engineering, Ekiti State University, P.M.B 5363, Ado-Ekiti, Nigeria E-mail: [email protected] Abstract Integration by parts is a well-known method of integrating product of two functions. If the integrand involves a polynomial of degree n and another function that can be integrated at least (n+1) times, then the solution to is guaranteed after n routine applications of integration by parts method. This paper presents a mathematical model for integrating product of two functions without going through the routine application of integration by parts at each stage of integration, thus saving a lot of computational time. Some examples were considered to illustrate the effectiveness of the model. The model is found to be appropriate as it gives the same result with the well-known integration by parts method. Keywords: Integrand; Mathematical model; Integration by parts; Differentiable functions 1. Introduction Integration by parts is a method used for integrating product of two functions particularly when either function is not a derivative of the other (Stroud & Dexter 2007). If and are two differentiable functions of x, then (1) is the formula for the derivative of product of two functions as we can see in Bhattacharyya (2009), Matthew & Alabi (2008), Dass (2009), Bajpai et al. (1981), Richmond (1972) and Oke (2003) , where . Rearranging equation (1) we have: Integrating equation (2) with respect to x, we have: 32
- 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 (3) is the formula for integration by parts as we can see in Kreyszig (1987), Thomas JR. & Finney (1984), Daniel (2000), Gupta (2009), Grossman (1985), and Goldstein et al. (2007). Let us assume that is a polynomial of degree n and let be a function of x that can be integrated at least (n+1) times. To evaluate using the integration by parts method, we will need n applications of the formula in (3) above before we can get a solution. In this paper, we derived a mathematical model for integrating product of two functions. In applying the model, we don’t need to go through the routine application of integration by parts each time we are integrating product of two functions. We only need to substitute the derivatives of and the integrals of in the newly derived model. 2. Materials and Methods Let and be two functions of , where is a polynomial of degree n and is a function of that can be integrated at least (n+1) times. Then: Since is a polynomial of degree n and can be integrated at least (n + 1) times, the procedure above will continue until the last stage where we now have: Putting all these in a more compact form, we have: 33
- 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 where is the jth derivative of and represents the (j + 1)th integral of with respect to . (10) above is the mathematical model for integrating product of two functions when the integrand involves a polynomial of degree n and another function which can be integrated at least (n+1) times. In order to apply the model to , we will find the derivatives of up to the nth derivative and the integrals of up to the (n + 1) th integral and substitute all these into the formula in (10) above to get our final result directly. 3. Computational Examples Example 1: Let us consider and , , and . From our notation: Similarly Substituting all these into the formula in (10) above we have our final result directly as: 34
- 4. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 where C is the constant of integration. Using the well-known integration by parts method to solve the problem, we have: . . where C is the constant of integration. This is the same with the result we got before. Example 2: Let us evaluate . and . , , and . From our notation: 35
- 5. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 Substituting all these into formula (10) above, we have our final result directly as: where C is the constant of integration. The integration by parts method, for this problem in example 2, gave us the same result after five routine applications. Example 3: Let us consider and and From our notation: . Putting all these in (10) above, we have our final result directly as: . . where C is the constant of integration. We also got the same result after two applications when we used the integration by parts method for the problem in example 3. Example 4: 36
- 6. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 Let us consider and , , and . From our notation: Putting all these in (10) above, we have our final result directly as: C is the constant of integration. We got the same result for this example when we used the integration by parts method after five routine applications. Conclusion In this paper, a mathematical model for integrating product of two functions was presented for an integrand involving a polynomial of degree n. The advantage of the model is in the computational time that is saved. We don’t need to go through the routine applications of integration by parts method each time we are integrating product of two functions. We only need to substitute the derivatives of up to the nth derivative and the integrals of up to the (n+1)th integral in the model. The mathematical model is found to be appropriate as it gives the same result with the integration by parts method. 37
- 7. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 References Bajpai, A.C., Calus, I.M., Fairley, J.A. & Walker, D. (1981), “Mathematics for Engineers and Scientists”, John Wiley and Sons. Bhattacharyya, B. (2009), “Mathematical Physics”, New Central Book Agency (P) Ltd. Daniel, A.A. (2000), “Application of Integral Calculus to Selected Science and Engineering Problems”, Ada Journal of Science, 8(2), 74-92. Dass, H.K. (2009), “Engineering Mathematics”, S. Chang & Company Ltd. Goldstein, L.J., Lay, D.C., Schneider, D.I. & Asmar, N.H. (2007), “Calculus and it Applications”, Pearson Education International. Grossman, S.I. (1985), “Applied Calculus”, Wadsworth Publishing Company. Gupta, B.D. (2009), “Mathematical Physics”, Vikas Publishing House PVT Ltd. Kreyszig, E. (1987) “Advanced Engineering Mathematics”, Wiley Eastern Limited. Matthew, D.A. & Alabi, P.O. (2008) “An Overview of Integration Techniques for Solving Engineering Problems”, Doe Journal of Science and Technology, 5(3), 80-98. Oke, M.O. (2003), “Calculus of One and Several Variables for Scientists and Engineers”, Petoa Educational Publishers. Richmond, A.E. (1972), “Calculus for Electronics”, Mc Graw-Hill Books Company. Stroud, K.A. & Dexter, J.B. (2007), “Engineering Mathematics”, Palgrave Macmillan Ltd. Thomas JR, G.B. & Finney, R.L. (1984) “Calculus and Analytic Geometry”, Addison-Wesley Publishing Company. Dr. M.O. OKE was born at Are – Ekiti in Nigeria on 16th April, 1969. He is a senior lecturer in the Department of Electrical and Electronic Engineering of Ekiti State University. He graduated with a B.Sc. degree in Industrial Mathematics from the University of Benin, Nigeria in 1991. He got a Postgraduate Diploma in Electrical and Electronic Engineering from the University of Ado-Ekiti, Nigeria and finished with a distinction in 2002.He had his M.Sc. degree in Mathematics from the University of Ilorin, Nigeria in 2005 and his Ph.D degree in Mathematics at the University of Ilorin, Nigeria this year (2012). Dr. M.O. Oke became a member of Nigerian Mathematical Society in 1992, Mathematical Association of Nigeria in 1994, National Association of Mathematical Physics in 1995 and American Mathematical Society in 2011. Dr. M.O. OKE specializes in Mathematical Modeling, Optimization and Numerical Analysis. 38
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