Question

Solve the following : `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))dx`

Answer 1

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)`

∴ I = `(4)/(3) (sqrt(2) - 1)`