How can I solve this tricky integral involving a radical and trig substitution?

  • Thread starter krusty the clown
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In summary, the problem is an integral with a radical in the denominator and an x^4 in the numerator. The conversation discusses different approaches to solving the integral, including possibly using a trig substitution or an incomplete beta function. However, after some discussion, it is determined that a simple substitution of u = x^2 and \frac{du}{2} = xdx will allow for an easier solution.
  • #1
krusty the clown
[tex] \int \frac {x}{\sqrt{3-x^4}}\mbox{dx}[/tex]

Basically I haven't had any luck with anything I have tried. It almost looks like a trig substitution, but I don't know how to deal with the x^4 under the radical. Also, I would rather have a hint as to where to start rather than have the whole problem done for me.


-Thanks, Erik
 
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  • #2
Deal it as [tex] (x^2)^2 [/tex]
 
  • #3
I think that's going to turn into an incomplete beta function.
 
  • #4
Tide said:
I think that's going to turn into an incomplete beta function.

I'm not sure what this means, but this problem is from calc II, and the text specifically says there are answers to all of the questions from the section this was taken from.

Cylclovenom said:
Deal it as [tex] (x^2)^2 [/tex]

Thanks. I am prety sure I can do this with trig substituion now, but if I can't I will let you know.


-Thanks, Erik
 
  • #5
Right! I didn't look at it long enough. It's going to be an arcsin with an x^2 in the argument.
 
  • #6
Read Tide's post he gave it away :smile:
No need for trigonometric substitution.

Hint!
[tex] u = x^2 [/tex]
[tex] \frac{du}{2}= xdx [/tex]
 

1. How do I identify when to use a trigonometric substitution for an integral involving a radical?

There are a few key signs to look for when deciding whether or not to use a trigonometric substitution for a tricky integral. First, check if the integral involves a radical with a variable inside, such as √(x^2 + a^2). This is a good indication that a trig substitution may be needed. Additionally, if the integral involves expressions like 1/(x^2 + a^2) or √(a^2 - x^2), a trigonometric substitution is likely necessary.

2. How do I choose the appropriate trig function to substitute?

To choose the appropriate trig function for substitution, you should consider the form of the integral and the trig identities that can be used to simplify it. For example, if the integral involves √(x^2 + a^2), a good substitution choice would be x = a tanθ. For 1/(x^2 + a^2), a substitution of x = a secθ may be more helpful. Additionally, it may be helpful to refer to a table of common trigonometric substitutions to guide your choice.

3. How do I solve for the new variable after making a trigonometric substitution?

After making a trigonometric substitution, you will need to solve for the new variable in terms of the original variable. This can be done by using the trigonometric identity associated with the chosen substitution. For example, if you substituted x = a tanθ, you can use the identity tanθ = x/a to solve for θ. Then, you can substitute this value back into the original integral and solve for the new variable.

4. What should I do if the integral involves both a radical and a trigonometric substitution?

In this case, you will need to use both the trigonometric substitution and a technique called integration by parts. First, make the trigonometric substitution as usual, and then use integration by parts to solve for the remaining radical term. You may also need to use trigonometric identities to simplify the integral before integrating.

5. How can I check my answer to ensure it is correct?

There are a few ways to check your answer for a tricky integral involving a radical and trig substitution. One method is to differentiate your answer and see if it matches the original integrand. You can also use a graphing calculator to graph both the original function and your answer and see if they intersect at all points. Additionally, you can use online integral calculators to compare your answer to the calculated solution.

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