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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