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Question
Evaluate the following definite integrals: `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))*dx`
Solution
Let I = `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))dx`
= `int_0^1 (1)/(sqrt(1 + x) + sqrt(x)) xx (sqrt(1 + x) - sqrt(x))/(sqrt(1 + x) - sqrt(x))dx`
= `int_0^1 (sqrt(1 + x) - sqrt(x))/((sqrt(1 + x))^2 - (sqrt(x)^2)`dx
= `int_0^1 (sqrt(1 + x) - sqrt(x))/(1 + x - x)dx`
= `int_0^1[(1 + x)^(1/2) - x^(1/2)]dx`
= `int_0^1 (1 + x)^(1/2)dx - int_0^1 x^(1/2)dx`
= `[((1 + x)^(1/2))/(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)`
= `(4sqrt2)/3 - 4/3`
∴ I = `(4)/(3) (sqrt(2) - 1)`
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