Question

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

Answer 1

Let I = `int_0^1 1/(1 + sqrt(x))*dx`
Put `sqrt(x)` = t
∴ x = t² and dx = 2t·dt
When x = 0, t = 0
When x = , t = 1

∴ I = `int_0^1 1/(1 + t)2t*dt`

= `2 int_0^1 t/(1 + t)*dt`

= `2 int_0^1 ((1 + t) - 1)/(1 + t)*dt`

= `2 int_0^1 (1 - 1/(1 + t))*dt`

= `2[t - log|1 + t|]_0^1`

= `2[1 - log2 - 0 + log1]`

= 2(1 - log 2)                             ...[∵ log 1 = 0]
= 2 – 2log 2
= 2 – log 4.