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Question
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
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Solution
Let I = `int 1/(sqrt(x + a) + sqrt(x + b))`
On multiplying the numerator and denominator by `sqrt(x + a) + sqrt(x + b)`
`= int 1/(sqrt(x + a) + sqrt(x + b)) xx (sqrt(x + a) - sqrt(x + b))/(sqrt(x + a) - sqrt(x + b))`
`= int (sqrt(x + a) - sqrt(x + b))/(sqrt(x + a) - sqrt(x + b))`
`= int (sqrt(x + a) - sqrt(x + b))/(a - b)` dx
`= 1/(a - b) int (x + a)^(1/2) dx - 1/(a - b) int (x + b)^(1/2)` dx
`= 1/(a - b) [2/3 (x + a)^(3/2) - 2/3 (x + b)^(3/2)] + C`
`= 2/(3(a - b))[(x + a)^(3/2) - (x + b)^(3/2)] + C`
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