Saturday, January 1, 2022

12 Days of Christmas Integrals: ∫ (x^2 - 4 ∙ x + 3) ÷ (x^2 - 6 ∙ x + 8) dx

 12 Days of Christmas Integrals:  ∫ (x^2 - 4 ∙ x + 3) ÷ (x^2 - 6 ∙ x + 8) dx



HAPPY NEW YEAR!!!!   HERE'S TO A WONDERFUL 2022!  May we make our future and present bright!  (We all could use a bright year, given how the last few years went)


On the Eighth day of Christmas Integrals, the integral featured today is...


∫ (x^2 - 4 ∙ x + 3) ÷ (x^2 - 6 ∙ x + 8) dx


By division:







= ∫ 1 + (2 ∙ x - 5) ÷ (x^2 - 6 ∙ x + 8) dx


Find the partial fraction of (2 ∙ x - 5) ÷ (x^2 - 6 ∙ x + 8) 


(2 ∙ x - 5) ÷ (x^2 - 6 ∙ x + 8) = A ÷ (x - 2) + B ÷ (x - 4)


2 ∙ x - 5 = A ∙ (x - 4) + B ∙ (x - 2)

2 ∙ x - 5 = A ∙ x - 4 ∙ A + B ∙ x - 2 ∙ B


which implies that:

-5 = -4 ∙ A - 2 ∙ B

2 = A + B


Solving for A and B:

A  = 1/2 and B = 3/2 


(2 ∙ x - 5) ÷ (x^2 - 6 ∙ x + 8) = (1/2) ÷ (x - 2) + (3/2) ÷ (x - 4)


= ∫ 1 + (1/2) ÷ (x - 2) + (3/2) ÷ (x - 4) dx


= x + 1/2 ∙ ln|x - 2| + 3/2 ∙ ln|x - 4| + C


Eddie 


All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


  Casio fx-7000G vs Casio fx-CG 50: A Comparison of Generating Statistical Graphs Today’s blog entry is a comparison of how a hist...