Monday, March 21, 2022

March Calculus Madness Sweet Sixteen - Day 6: ∫ √a ÷ (√(x+a) - √x) dx

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Welcome to March Calculus Madness!


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∫ √a ÷ (√(x+a) - √x) dx



Simplify:

√a ÷ (√(x+a) - √x)


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


√a ÷ (√(x+a) - √x)

= √a ÷ (√(x+a) - √x) * (√(x+a) + √x) ÷ (√(x+a) + √x) 

= √a * (√(x+a) + √x) ÷ (x + a - x)

= 1 ÷ √a * (√(x+a) + √x)


Then:

∫ √a ÷ (√(x+a) + √x) dx

=  ∫ 1 ÷ √a * (√(x+a) + √x) dx

= a^(-1/2) * ∫ (x+a)^(1/2) + x^(1/2) dx

= 2 ÷ (3 * a^(1/2)) * ((x + a)^(3/2) + x^(3/2)) + C

= 2 ÷ (3 * a^0.5) * ((x + a)^1.5 + x^1.5) + C



Questions of preference:


Which is your preferred notation:


(1)  √x,  x^(1/2), or x^0.5?


(2)  x^(3/2) or x^1.5?  


Let us know in the comments.



Eddie 



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