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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