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 


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