Question

Evaluate the definite integral:

`int_0^1 dx/(sqrt(1+x) - sqrtx)`

Answer 1

Let  I = `int_0^1 dx/(sqrt(1 + x) - sqrtx)`

On multiplying the numerator and denominator by `sqrt(1 + x) - sqrtx`

I = int_0^1 (sqrt(1 + x) - sqrtx)/(1 + x - x)  dx`

`= int_0^1 (sqrt(1 + x) - sqrtx)  dx`

`= int_0^1 sqrt(1 + x) dx + int_0^1 sqrtx  dx`

`= [2/3 (1 + x)^(3//2)]_0^1 + [2/3 x^(3//2)]_0^1`

`= 2/3 (2^(3//2) - 1) + 2/3 [1 - 0]`

`= 2/3 * 2^(3//2) - 2/3 + 2/3`

`= 2/3 * 2^(3//2)`

`= 2/3 * 2sqrt2`

`= (4sqrt2)/3`