Question

How do you evaluate the definite integral `int 2/sqrt(1+x)` from `[0,1]`?

Answer 1

`4(sqrt2-1)`.

Explanation:

Let `1+x=t^2 rArr dx=2tdt`

Also, `x=0 rArr t^2=1 rArr t=1, &, x=1 rArr t=sqrt2`. Hence,

`I=int_0^1 2/sqrt(1+x)dx=int_1^sqrt2 2/sqrt(t^2)*2tdt=4int_1^sqrt2 dt`

`=4[t]_1^sqrt2`

`=4(sqrt2-1)`.