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

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In summary, the conversation is about a difficult integral problem involving a square root and a fraction in the argument. The person is struggling to solve it and has tried a few different approaches, including using an online calculator. Another person suggests a substitution of u=sqrt(2)x^(3/4) which leads to a simpler integral. The first person clarifies that they did not cheat and is just self-learning.
  • #1
Stratosphere
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Homework Statement


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


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
 
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  • #2
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?
 
  • #3
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.
 
  • #4
Well, just glancing at the integral it seems like the substitution [itex]u=\sqrt{\theta}[/itex] is a decent place to start...have you tried that? If so, how far did you get with it?
 
  • #5
Stratosphere said:
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.
 

1. What is the best approach to solving this tricky integral?

The best approach to solving this integral is to use the substitution method. Let u = √(θ + ½θ−½) and then solve for θ in terms of u. This will simplify the integral and make it easier to solve.

2. Can this integral be solved using basic integration techniques?

Yes, this integral can be solved using basic integration techniques such as substitution and integration by parts. However, it may require multiple steps and careful manipulation of the expression to arrive at the final solution.

3. Is there a specific value for θ that can be substituted to make this integral easier?

Unfortunately, there is no specific value for θ that can be substituted to simplify this integral. However, using the substitution method as mentioned above can make the integral more manageable.

4. Can this integral be solved using numerical methods?

Yes, this integral can also be solved using numerical methods such as Simpson's rule or the trapezoidal rule. However, these methods may not provide an exact solution but rather an approximation.

5. What are some common mistakes to avoid when solving this integral?

Some common mistakes to avoid when solving this integral include forgetting to use the chain rule in the substitution method and not carefully simplifying the expression before integrating. It is also important to check for any algebraic mistakes before arriving at the final solution.

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