Tuesday, January 4, 2022

12 Days of Christmas Integrals: ∫ (a ∙ b) ÷ (a^2 ∙ x^2 + b^2) dx; a,b are constants

 12 Days of Christmas Integrals:  ∫ (a ∙ b) ÷ (a^2 ∙ x^2 + b^2) dx; a, b are constants


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


∫ (a ∙ b) ÷ (a^2 ∙ x^2 + b^2) dx; a, b are constants


A rather simple integral:


∫ (a ∙ b) ÷ (a^2 ∙ x^2 + b^2) dx


= a ∙ b ∙ ∫ 1÷ (a^2 ∙ x^2 + b^2) dx


= a ∙ b ∙ ∫ 1÷ (b^2 ∙ (a^2/b^2 ∙ x^2 + 1)) dx


= a/b ∙ ∫ 1÷ (a^2/b^2 ∙ x^2 + 1) dx


= ∫ (a/b) ÷ ((a/b)^2 ∙ x^2 + 1) dx


= arctan(a/b ∙ x) + C



Eddie 



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