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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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