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Question
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
Solution
Let I = `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
= `int_0^1 1/(sqrt(1 + x) + sqrt(x)) xx (sqrt(1 + x) - sqrt(x))/(sqrt(1 + x) - sqrt(x)) "d"x`
= `int_0^1 (sqrt(1 + x) - sqrt(x))/((sqrt(1 + x))^2 - (sqrt(x))^2) "d"x`
= `int_0^1 (sqrt(1 + x) - sqrt(x))/(1 + x - x) "d"x`
= `int_0^1 [(1 + x)^(1/2) - x^(1/2)] "d"x`
= `int_0^1 (1 + x)^(1/2) "d"x - int_0^1 x^(1/2) "d"x`
= `[(1 + x)^(3/2)/(3/2)]_0^1 - [(x^(3/2))/(3/2)]_0^1`
= `2/3 [(2)^(3/2) - (1)^(3/2)] - 2/3 [(1)^(3/2) - 0]`
= `2/3(2sqrt(2) - 1) - 2/3(1)`
= `(4sqrt(2))/3 - 2/3 - 2/3`
∴ I = `4/3 (sqrt(2) - 1)`
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