Journal of Xi’an Shiyou University, Natural Science Edition ISSN : 1673-064X THE ANTI-DERIVATIVE FUNCTIONS KNOWN AS BANGABANDHU FUNCTIONS Probir Roy Associate Professor, Dept. of Mathematics, Murarichand (M.C) College, Sylhet-3100, Bangladesh ( Affiliated to Bangladesh National University ) Dedication: This paper is dedicated to the greatest Bengali of the millennium, the great architect of independent Bangladesh, the father of the Bengali nation, Bangabandhu Sheikh Mujibur Rahman, and the functions are named after him. ABSTRACT: We can evaluate every definite integral using the limit or summation or the Trapezoidal rule and any other established rules. With the help of anti-derivative and the Fundamental Theorem of Calculus, we can also compute the value of various definite integrals. But every definite integral can’t be evaluated by anti-derivative because every function has no anti- derivative formula as a function of the real variable. In that case, the function should be integrable within the interval. An attempt has been made here to establish general anti-derivative formulae named “Bangabandhu functions,’’ for such types of particular functions which have no elementary functions for anti-derivative in this paper. From these formulae, we can easily calculate the area bounded by the graph of the given function and other boundaries up to certain decimal places. Keywords : Integral, anti-derivative, generalized, Gaussian, approximated 1. INTRODUCTION The purpose of integration is to determine the area of plane surfaces, area of curved surfaces, and the volume of solids, which sometimes can be calculated easily with the help of anti-derivatives and the Fundamental Theorem of Calculus concerning some conditions. Archimedes, the great scientist of antiquity, solved some problems in determining the area and volume of solids in the 3rd century BC, which are considered the basis of Integral Calculus. Later, in the hands of world- famous scientists like Isaac Newton, Gottfried Wilhelm Leibniz, Bernhard Riemann, Integral Calculus has become an essential modern tool of science today. If we can memorize a formula that is derivative of a function or can find an anti-derivative to that function by applying techniques, then we can easily find the respective area bounded by that function. And if the function intertwined with the field is not integrable, then the surprisingly Fundamental Theorem of Calculus, cannot be used to determine the area. However, it is possible to find the area by more than one established rule (Limit, Summation, the Trapezoidal, the Simpson, etc.) Our main objective of this paper is to establish general anti-derivative formulae for the functions like G(x) = 푥푛 −푥푛 푏 푥푛 푏 −푥푛 푒 , 푒 , 푛 ∈ N and how to solve ∫푎 푒 푑푥, ∫푎 푒 푑푥 through a simple method (using expansion of a function in a series) without using error functions erf(x), erfi(x), erfc(x), etc., and Gamma function. [2, 5] http://xisdxjxsu.asia VOLUME 17 ISSUE 06 212-224 Journal of Xi’an Shiyou University, Natural Science Edition ISSN : 1673-064X 2. RELATED WORKS In the past, Toyesh Prakash Sharma [8], in most of the books [1] , [ see page 712 [7] ] used a similar process to evaluate the indefinite and definite integral. Here has been tried to express the work into a simple form using expansion of a function, summation of series, and mathematical induction rule along with the Fundamental Theorem of Calculus. 3. MATHEMATICAL BACKGROUND 3.1 Anti-derivative: A function F(x) is said to be anti-derivative of another function G(x) in an interval if F′(푥) = G(푥) holds and it is generally denoted by ∫ G(푥)푑푥 = F(푥). Again, since 푑 {F(푥) + c} = F′(푥) for any real value of c, so the anti-derivative or indefinite integral of G(x) is 푑푥 ∫ G(푥)푑푥 = F(푥) + c , where c is an arbitrary constant which is called constant of integration. Example 1:Let G(푥) = 2푥 and F(푥) = 푥2, now F′(푥) = 2푥 = G(푥) so, ∫ 2푥 푑푥 = 푥2 + c 3.2 Summation of series: If an infinite series is S∞ = a + (a+d) + (a+2d) + ……………∞, then the m-th term of the series is denoted by {a + (m-1).d} and the summation of the series is ∞ S푚 = ∑푚=1{푎 + (푚 − 1). 푑} 3.3 Fundamental Theorem of Calculus: If 푦 = G(푥) is an integrable function in the closed interval [a, b] and the anti-derivative of G(푥) in the open interval (a, b) is F(푥) + c, then the area or 푏 ( ) definite integral ∫푎 G(푥) 푑푥 = F 푏 − F(푎) 1 푥 푥 1 ( 1 ) 0 Example 2: ∫0 푒 푑푥 = [푒 + c]0 = 푒 + c − (푒 + c) = 푒 − 1 4. MATHEMATICAL INDUCTIONS 푥 4.1 Induction1: Let 퐵1(x) = ∫ 푒 푑푥 푥 푥2 푥3 푥4 푥5 푥6 We know that 푒푥 = 1 + + + + + + + ⋯ … … … … … … ∞ 1! 2! 3! 4! 5! 6! 푥 푥2 푥3 푥4 푥5 푥6 ∴ 퐵 (x) = ∫ 푒푥 푑푥 = ∫(1 + + + + + + + ⋯ … … … … … … ∞) 푑푥 1 1! 2! 3! 4! 5! 6! 푥2 푥3 푥4 푥5 푥6 푥7 = (푥 + + + + + + + ⋯ … … … … … … ∞) + c 2.1! 3.2! 4.3! 5.4! 6.5! 7.6! 푥1 푥2 푥3 푥4 푥5 푥6 푥7 = ( + + + + + + + ⋯ … … … … … … ∞) + c 1.0! 2.1! 3.2! 4.3! 5.4! 6.5! 7.6! 푥푚 = ∑∞ + c (3.2) 푚=1 푚.(푚−1)! 푥푚 Therefore, the anti-derivative of 푒푥 i.e., 퐵 (x) = ∫ 푒푥 푑푥 = ∑∞ + c 1 푚=1 푚.(푚−1)! Similarly, we can prove that the anti-derivative of 푒−푥 (−1)푚−1 푥푚 i.e., 퐵 (x) = ∫ 푒−푥 푑푥 = ∑∞ + c 1∗ 푚=1 푚.(푚−1)! 푥1 푥2 푥3 푥4 푥5 푥6 푥7 Note: 퐵 (x) = ( + + + + + + + ⋯ … … … … … … ∞) + c 1 1.0! 2.1! 3.2! 4.3! 5.4! 6.5! 7.6! 푥1 푥2 푥3 푥4 푥5 푥6 푥7 = (1 + + + + + + + + ⋯ … … … … … … ∞) + (c − 1) 1.0! 2.1! 3.2! 4.3! 5.4! 6.5! 7.6! 푥 푥2 푥3 푥4 푥5 푥6 = (1 + + + + + + + ⋯ … … … … … … ∞) + (c − 1) 1! 2! 3! 4! 5! 6! http://xisdxjxsu.asia VOLUME 17 ISSUE 06 212-224 Journal of Xi’an Shiyou University, Natural Science Edition ISSN : 1673-064X 푥 푥2 푥3 푥4 푥5 푥6 = (1 + + + + + + + ⋯ … … … … … … ∞) + c [Since c is an arbitrary constant of 1! 2! 3! 4! 5! 6! integration] = 푒푥 + c ∴ ∫ 푒푥푑푥 = 푒푥 + c, Therefore, there is no contradiction with the usual anti-derivative of 푒푥 푥2 4.2 Induction2: Let 퐵2(x) = ∫ 푒 푑푥 푥 푥2 푥3 푥4 푥5 푥6 We know that 푒푥 = 1 + + + + + + + ⋯ … … … … … … ∞ 1! 2! 3! 4! 5! 6! 2 푥2 푥4 푥6 푥8 푥10 푥12 ∴ 퐵 (x) = ∫ 푒푥 푑푥 = ∫(1 + + + + + + + ⋯ … … … … … … ∞) 푑푥 2 1! 2! 3! 4! 5! 6! 푥3 푥5 푥7 푥9 푥11 푥13 = (푥 + + + + + + + ⋯ … … … … … … ∞) + c 3.1! 5.2! 7.3! 9.4! 11.5! 13.6! 푥1 푥3 푥5 푥7 푥9 푥11 푥13 = ( + + + + + + + ⋯ … … … … … … ∞) + c 1.0! 3.1! 5.2! 7.3! 9.4! 11.5! 13.6! 푥2푚−1 = ∑∞ + c (3.2) 푚=1 (2푚−1).(푚−1)! 2 2 푥2푚−1 Therefore, the anti-derivative of 푒푥 i.e., 퐵 (x) = ∫ 푒푥 푑푥 = ∑∞ + c which is free 2 푚=1 (2푚−1).(푚−1)! of erfi(x) [5] 2 Similarly, we can prove that the anti-derivative of 푒−푥 2 (−1)푚−1 푥2푚−1 i.e., B (x) = ∫ 푒−푥 푑푥 = ∑∞ + c which is free of erf(x) [5] 2∗ 푚=1 (2푚−1).(푚−1)! 푥3 4.3 Induction3: Let 퐵3(x) = ∫ 푒 푑푥 푥 푥2 푥3 푥4 푥5 푥6 We know that 푒푥 = 1 + + + + + + + ⋯ … … … … … … ∞ 1! 2! 3! 4! 5! 6! 3 푥3 푥6 푥9 푥12 푥15 푥18 ∴ 퐵 (x) = ∫ 푒푥 푑푥 =∫(1 + + + + + + + ⋯ … … … … … … ∞) 푑푥 3 1! 2! 3! 4! 5! 6! 푥4 푥7 푥10 푥13 푥16 푥19 = (푥 + + + + + + + ⋯ … … … … … … ∞) + c 4.1! 7.2! 10.3! 13.4! 16.5! 19.6! 푥1 푥4 푥7 푥10 푥13 푥16 푥19 = ( + + + + + + + ⋯ … … … … … … ∞) + c 1.0! 4.1! 7.2! 10.3! 13.4! 16.5! 19.6! 푥3푚−2 = ∑∞ + c (3.2) 푚=1 (3푚−2).(푚−1)! 3 3 푥3푚−2 Therefore, the anti-derivative of 푒푥 i.e., 퐵 (x) = ∫ 푒푥 푑푥 = ∑∞ + c 3 푚=1 (3푚−2).(푚−1)! 3 Similarly, we can prove that the anti-derivative of 푒−푥 3 (−1)푚−1 푥3푚−2 i.e., 퐵 (x) = ∫ 푒−푥 푑푥 = ∑∞ + c 3∗ 푚=1 (3푚−2).(푚−1)! 푥4 4.4 Induction4: Let 퐵4(x) = ∫ 푒 푑푥 푥 푥2 푥3 푥4 푥5 푥6 We know that 푒푥 = 1 + + + + + + + ⋯ … … … … … … ∞ 1! 2! 3! 4! 5! 6! 4 푥4 푥8 푥12 푥16 푥20 푥24 ∴ 퐵 (x) = ∫ 푒푥 푑푥 = ∫(1 + + + + + + + ⋯ … … … … … … ∞) 푑푥 4 1! 2! 3! 4! 5! 6! 푥5 푥9 푥13 푥17 푥21 푥25 = (푥 + + + + + + + ⋯ … … … … … … ∞) + c 5.1! 9.2! 13.3! 17.4! 21.5! 25.6! 푥1 푥5 푥9 푥13 푥17 푥21 푥25 = ( + + + + + + + ⋯ … … … … … … ∞) + c 1.0! 5.1! 9.2! 13.3! 17.4! 21.5! 25.6! 푥4푚−3 = ∑∞ + c (3.2) 푚=1 (4푚−3).(푚−1)! http://xisdxjxsu.asia VOLUME 17 ISSUE 06 212-224 Journal of Xi’an Shiyou University, Natural Science Edition ISSN : 1673-064X 4 4 푥4푚−3 Therefore, the anti-derivative of 푒푥 i.e., 퐵 (x) = ∫ 푒푥 푑푥 = ∑∞ + c 4 푚=1 (4푚−3).(푚−1)! 4 Similarly, we can prove that the anti-derivative of 푒−푥 4 (−1)푚−1 푥4푚−3 i.e., 퐵 (x) = ∫ 푒−푥 푑푥 = ∑∞ + c 4∗ 푚=1 (4푚−3).(푚−1)! 5. MAIN RESULT From the above results, we can decide with the help of the mathematical induction rule that The anti-derivative Bangabandhu functions as follows: 푛 푥푛.푚−(푛−1) 퐵 (x) = ∫ 푒푥 푑푥 = ∑∞ + c, where 푛 ∈ N 푛 푚=1 {(푛.푚−(푛−1)}.(푚−1)! And 푛 (−1)푚−1 푥푛.푚−(푛−1) 퐵 (x) = ∫ 푒− 푥 푑푥 = ∑∞ + c, where 푛 ∈ N [Generalized error functions [2]] 푛∗ 푚=1 {(푛.푚−(푛−1)}.(푚−1)! ퟏ 풙풏 ퟏ −풙풏 6.