Tuesday, December 28, 2021

12 Days of Christmas Integrals: ∫ x^2 ÷ (1 + x^2) dx

 12 Days of Christmas Integrals:  ∫ x^2 ÷ (1 + x^2) dx


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


∫ x^2 ÷ (1 + x^2) dx


Time for a little algebraic manipulation:  add and subtract 1 ÷ (1 + x^2) 


= ∫ x^2 ÷ (1 + x^2) + 1 ÷ (1 + x^2) - 1 ÷ (1 + x^2)   dx


= ∫ (x^2 + 1) ÷ (1 + x^2) - 1 ÷ (1 + x^2)   dx


= ∫ (1 + x^2) ÷ (1 + x^2) - 1 ÷ (1 + x^2)   dx


= ∫ 1 - 1 ÷ (1 + x^2)   dx


= x - arctan(x) + C


Eddie 


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