Indefinite integral and proving convergence

In summary, the conversation discusses a problem involving ∫(x*sin2(x))/(x3-1) over the terminals b= ∞ and a = 2, and the use of various rules such as the p-test, ratio test, and squeeze test to determine its convergence or divergence. It is suggested to use the comparison test and compare the integral to 1/x3. A theorem is mentioned that states if ∫|f(x)| converges, then ∫f(x) also converges, and this is applied to the given problem. The process of removing absolute values is also discussed, and it is suggested to start with a partial fraction expansion and
  • #1
SteliosVas
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Homework Statement



okay so the equation goes:

∫(x*sin2(x))/(x3-1) over the terminals:
b= ∞ and a = 2

Homework Equations



Various rules applying to the convergence or divergence of integrals such as the p-test, ratio test, squeeze test etc

The Attempt at a Solution



Okay so I have tried factorising the bottom, I have tried the squeeze test and the ratio test to no success..

would method should I be applying, because I am unsure how to split up this integral...

Thanks
 
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  • #2
Here's a hint: first notice that [itex]\sin^2 x \leq 1[/itex] for all [itex]x[/itex]. Then you can compare your integral with another one whose integrand is always greater. You can then prove the convergence of that integral.
 
  • #3
Therefore using the comparison test should i compare this integral to 1/x3 over the same interval?
 
  • #4
Lets apply a useful theorem. If ##\int_a^b |f(x)| \space dx## converges, then ##\int_a^b f(x) \space dx## converges. Note the converse of this theorem is not true. So you should always determine the convergence of ##\int_a^b |f(x)| \space dx## to determine if ##\int_a^b f(x) \space dx## converges.

Applying this theorem to your given problem:

$$\int_{2}^{\infty} \left| \frac{x \sin^2(x)}{x^3 - 1} \right| \space dx = \int_{2}^{\infty} \frac{|x| |\sin^2(x)|}{|x^3 - 1|} \space dx \leq \int_{2}^{\infty} \frac{|x|}{|x - 1| |x^2 + x + 1|} \space dx$$

For ##x \in [2, \infty)##, we can remove the absolute values in the expression because all of the terms would be positive anyway. So the new question is, does this integral converge:

$$\int_{2}^{\infty} \frac{x}{(x - 1) (x^2 + x + 1)} \space dx$$

Hint: Start with a partial fraction expansion, and then complete the square.
 
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  • #5
Right now I understand. So the absolute of f(x) is always said to converge but the the opposite isn't always true ?

Right do we are performing a comparison test to x/(x^3-1)
 

1. What is an indefinite integral?

An indefinite integral is the reverse process of differentiation. It is a mathematical operation that finds the most general antiderivative of a given function. The output of an indefinite integral is a function, rather than a specific number.

2. How is an indefinite integral different from a definite integral?

A definite integral has specific limits of integration and gives a numerical value as the output. An indefinite integral has no limits of integration and gives a function as the output.

3. What is the significance of proving the convergence of an indefinite integral?

Proving the convergence of an indefinite integral is important in determining whether the integral is well-defined and has a finite value. It also helps in finding the area under a curve, which has various applications in mathematics and physics.

4. What methods are used to prove the convergence of an indefinite integral?

There are various methods to prove the convergence of an indefinite integral, such as the comparison test, integral test, and limit comparison test. These tests compare the given integral to a known convergent or divergent series to determine its convergence.

5. Can an indefinite integral be divergent?

Yes, an indefinite integral can be divergent if the given function is not well-behaved or if the limits of integration are infinite. In such cases, the integral does not have a finite value and is considered divergent.

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