The Five Columns Rule in Solving Definite Integration by Parts Through Transformation of Integral Limits

Abstract

This research introduces alternative method in solving definite integration by parts. The integral is solved using an algorithm of the tabular integration by parts through transformation of integral limits. The results revealed that the integration by parts formula changed to a new form after transformation. The implication of this method revealed that the final solution between this technique and standard techniques in calculus text books are exactly the same. The problem in this method is the procedures becomes longer than usual, therefore, the authors has derived the five columns rule to simplify the problem. The explanation of each columns are as follows: the first column writes the positive and negative signs alternately, the second column is f(x), and then successive its derivative that lies below it repeatedly such that the derivative equal to zero, the third column is g’(x), and then its integral lies below it, the fourth column writes the lower limit (x = a) and the upper limit (x = b), and the fifth column is the transformation of integral limits, i.e. the lower double limit {u(a)v(a)} and upper double limit {u(b)v(b)} on the integral respectively.