Calculating Improper Integral of x^3/(x^4-3)^1/2

In summary, the conversation discusses the process of calculating an improper integral and the resulting expressions when taking the limits. It is concluded that the integral in question is divergent, possibly due to an algebra error, and that the second expression is a constant.
  • #1
FancyNut
113
0
I stopped at the last step while calculating this improper integral:

integral of x^3\ ( x^4 - 3)^1\2 with limits from 1 to infinity...

that's x cubed over the square root of x raised to 4 minus 3...



after replacing infinity with b and taking the limits it seems that I have to take limits of two expressions one that goes to infinity which is ( (b^4 - 3)^1\2) / 8 but the other is undefined since I have a negative 2 inside a square root. The second expression is the square root of negative two all over 8.


Thanks for any help XD
 
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  • #2
Just to make sure, that's:
[tex]\int_1^{\infty} \frac{x^3}{\sqrt{x^4-3}} dx[/tex]
Which is divergent.

Perhaps there was an algebra error leading up to this?
 
  • #3
Not only is it divergent, but it makes no sense outside the field of complex analysis..
 
  • #4
Yeah that's it.

I end up with two expressions and when taking the limit of the first it's infinity but the second has a negative inside the sqaure root. I guessed it's divergent since whatever mistake I did doesn't change (I think..) that the second expression is a constant anyway... I guess. :p

Thanks dudes. ^_^
 

1. How do I calculate the improper integral of x^3/(x^4-3)^1/2?

To calculate this improper integral, you will need to use a technique called partial fractions. First, factor the denominator into its irreducible factors. Then, set up a system of equations using the partial fractions formula and solve for the unknown coefficients. Once you have the partial fraction decomposition, you can integrate each term separately and add them together to get the solution.

2. What is the domain of the function x^3/(x^4-3)^1/2?

The domain of this function is all real numbers except the values that would make the denominator equal to zero. In this case, the denominator would be equal to zero when x is equal to ±√3, so the domain is (-∞, -√3) ∪ (-√3, ∞).

3. Can I use substitution to solve this improper integral?

No, substitution cannot be used to solve this improper integral because the function is not in the form of u-substitution. The exponent of x in the numerator is not one less than the exponent in the denominator, which is required for u-substitution to work.

4. What is the significance of the exponent of x in the numerator and denominator?

The exponent of x in the numerator and denominator plays a crucial role in determining the convergence or divergence of this improper integral. If the exponent in the numerator is greater than the exponent in the denominator, the integral will diverge. If the exponent in the numerator is less than the exponent in the denominator, the integral will converge. In this case, the exponents are equal, so further analysis is needed to determine convergence or divergence.

5. Can I use a graphing calculator to find the value of this improper integral?

Yes, you can use a graphing calculator to approximate the value of this improper integral. However, it is important to note that graphing calculators use numerical methods to approximate integrals, so the result may not be exact. It is always best to solve the integral analytically for an exact solution.

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