How Can You Solve the Integral of x sqrt(x/(x-1)) Without a Trig Substitution?

[Sidthewall]

Homework Statement

integral of x *sqrt( x/(x-1))

Homework Equations

The Attempt at a Solution

I honestly don't know how to approach this i don't see any type of trig substituions whatsover, I really need some kind of lead and an explnation y that method works

 

[gabbagabbahey]

Homework Statement

integral of x *sqrt( x/(x-1))

Homework Equations

The Attempt at a Solution

I honestly don't know how to approach this i don't see any type of trig substituions whatsover, I really need some kind of lead and an explnation y that method works

Hmmm... I guess I'd start by making the substitution u=\frac{x}{x-1}=1+\frac{1}{x-1} in order to get rid of the junk inside the square root. You'll probably find that doing this will allow you to split the integral into two easier integrals since x=1+\frac{1}{u-1}

 

[Mark44]

Homework Statement

integral of x *sqrt( x/(x-1))

Homework Equations

The Attempt at a Solution

I honestly don't know how to approach this i don't see any type of trig substituions whatsover, I really need some kind of lead and an explnation y that method works

I haven't carried this all the way through, but I think it will work
\int x \sqrt{\frac{x}{x - 1}}dx = \int \frac{x^{3/2}}{\sqrt{x - 1}}dx

I believe that an ordinary substitution will work.
Let u = sqrt(x - 1).

 

[mmmboh]

Using that substitution you are going to end up having to do a trig. substitution, and the integral will be of (sec v)^5 which is very difficult. Are you sure this is the integral you have? it seems a bit complicated...this is the answer http://www.wolframalpha.com/input/?i=integral+of+x%28x%2F%28x-1%29%29^0.5.

 

[vela]

You can integrate secⁿ θ using integration by parts, letting u=sec^n-2 θ and dv=sec² θ dθ. For sec⁵ θ, you'll have to do it twice. It's a bit tedious but straightforward.

 

[gabbagabbahey]

You don't actually need a trig sub. You can also use Partial Fraction Decomposition if you find that easier.