Solve Tricky Integral: \int \sqrt{\theta+\frac{1}{2}\theta^{-1/2}}d\theta

[Stratosphere]

Homework Statement

I can't seem to figure out what I need to do in order to solve this one.
$$\int \sqrt{\theta+\frac{1}{2}\theta^{-1/2}}d\theta$$

Homework Equations

The Attempt at a Solution

I haven't the slightest clue as to what technique to use, it doesn't look to be a u substitution problem though (I could be wrong).
I typed it in on an online calculator and I got some really strange expression.

http://integrals.wolfram.com/index.jsp?expr=%28x%2B1%2F2x^%28-1%2F2%29%29^%281%2F2%29&random=false

 

[Dick]

It is a substitution problem. Since you already cheated and looked at wolfram you may as well use that cheat. Look at the argument of the ArcSinh. That suggests if you substitute u=sqrt(2)x^(3/4) you should be able to reduce it to something you can handle. In fact, once you hack through the radicals you should be able to show that aside from the numeric factors, it becomes sqrt(u^2+1)du. Can you integrate that?

 

[Stratosphere]

I don't want to use wolfram, I just used it because I couldn't figure it out. I didn't cheat on homework or anything I am self learning it. So what would be the first step starting from the original equation.

 

[gabbagabbahey]

Well, just glancing at the integral it seems like the substitution $u=\sqrt{\theta}$ is a decent place to start...have you tried that? If so, how far did you get with it?

 

[Dick]

I don't want to use wolfram, I just used it because I couldn't figure it out. I didn't cheat on homework or anything I am self learning it. So what would be the first step starting from the original equation.

I'm exaggerating on the 'cheating' aspect. But I already suggested an initial substitution in my last post.