Question

Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x))  "d"x`

Answer 1

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)`