Sunday, January 2, 2022

12 Days of Christmas Integrals: ∫ 1 ÷ (√(x + a) - √(x)) dx, a is a constant

 12 Days of Christmas Integrals:  ∫ 1 ÷ (√(x + a) - √(x)) dx, a is a constant


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


∫ 1 ÷ (√(x + a) - √(x)) dx, a is a constant  



Multiply by (√(x+a) + √(x)) ÷ (√(x+a) + √(x)):


= ∫ 1 ÷ (√(x + a) - √(x)) ∙ (√(x+a) + √(x)) ÷ (√(x+a) + √(x)) dx


= ∫ (√(x+a) + √(x)) ÷ (x + a - x) dx


= ∫ (√(x+a) + √(x)) ÷ (a) dx


= ∫ √(x + a) ÷ a + √(x) ÷ a dx


= 2/3 ∙ (x + a)^(3/2) + 2/3 ∙ x^(3/2) ÷ a


= 2 ÷ (3 ∙ a) ∙ ((x + a)^(3/2) + x^(3/2))


Eddie 


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