Expert: Paul Klarreich - 2/15/2010

Question
Hi. this is the problem
int(x^(-1/2) * (1 + x^(1/2))^(-1/2) dx
attempt:
u=x^(1/2)
du/dx = 1/2  x^(-1/2)
du= 1/2  x^(-1/2) dx
2 du = x^(-1/2) dx
by substituting to the original
int(2(1+u)^(-1/2)du)
here I am stuck

Answer
Questioner: hamad
Country: Kuwait
Category: Calculus
Private: No
Subject: integration
Question: Hi. this is the problem
int(x^(-1/2) * (1 + x^(1/2))^(-1/2) dx
attempt:
u=x^(1/2)
du/dx = 1/2  x^(-1/2)
du= 1/2  x^(-1/2) dx
2 du = x^(-1/2) dx
by substituting to the original
int(2(1+u)^(-1/2)du)
here I am stuck
.......................................
You have:

int(x^(-1/2) * (1 + x^(1/2))^(-1/2) dx

which is:

         dx
--------------------------
sqrt(x) sqrt(1 + sqrt(x))

let u = sqrt(x),  then  u^2 = x  and  2u du = dx

So you will have

   2 u du
-------------- =
u sqrt(1 + u)

   2 du
-------------
sqrt(1 + u)

Is this what you have?  It does look like yours.

So, what's the problem?  Now do  v = 1 + u,  dv = du, and you have:

 2 dv
----------
sqrt(v)


=  2 v^-1/2 dv

and from here you should be OK.