# How to integrate sqrt((ax+b)/x) dx (square root of linear fractional function)

1. May 27, 2012

### Holali

Hi,
simple question, but difficult to find an answer for me

How to integrate sqrt((ax+b)/x) dx ???
a,b constants and x variable

if it matters, I would be happy if you could solve it just for both a,b >0

Thanks

2. May 27, 2012

### algebrat

Let $u=\frac{ax+b}{x}$. We're keeping our fingers crossed, there's no other common methods that apply, so this looked like most natural thing to do.

The expression will be a little simpler, but now we try a more common method, which is let $v=\sqrt{u}$. To find $du$, we usually consider $v^2=u$, to simplify things. I think you will end up with something that you know how to integrate.

3. May 29, 2012

### JJacquelin

Let x = (b/a)(sinh(t))²

4. May 30, 2012

### rhythmiccycle

go here:

integrals.wolfram.com/index.jsp

and enter

sqrt[(a*x+b)/x]

into the text box