Friday, December 31, 2021

12 Days of Christmas Integrals: ∫ x ∙ (ln(x))^2 dx

12 Days of Christmas Integrals:  ∫ x ∙ (ln(x))^2 dx


NEW YEARS EVE!!!!


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


∫ x ∙ (ln(x))^2 dx


Sounds like a job for integration by parts!


∫ x ∙ (ln(x))^2 dx


u = (ln(x))^2 

du = 2 ∙ ln(x) ∙ 1/x dx

dv = x dx

v = x^2/2


= x^2/2 ∙ (ln(x))^2 - ∫ 2 ∙ ln(x) ∙ 1/x  ∙ x^2/2 dx


= x^2/2 ∙ (ln(x))^2 - ∫ x ∙ ln(x) dx


u  = ln(x)

du = 1/x dx

dv = x dx

v = x^2/2


= x^2/2 ∙ (ln(x))^2 - x^2/2 ∙ ln(x) + ∫ x/2 dx


= x^2/2 ∙ (ln(x))^2 - x^2/2 ∙ ln(x) + x^2/4 + C


Eddie 


All original content copyright, © 2011-2021.  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. 


TI-84 Plus CE: Dead Reckoning

TI-84 Plus CE:  Dead Reckoning Going Where You Want To Go Today's program is Dead Reckoning is a widely used  process that calculates th...