## 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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