Approaching Infinity: Solving Improper Integrals with Calc II Techniques

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In summary, the conversation is about finding the improper integral of x/(x^2+2)(x^2+2) dx from 0 to infinity. The poster attempted integrating by parts and partial fractions, but neither method seemed to work. Another user suggested using the substitution u=x^2+2, which simplifies the integral and makes it solvable. It turns out that the solution was much simpler than the original poster had thought.
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demersal
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Homework Statement


[tex]\int\frac{x}{(x^2+2)(x^2+2)}[/tex] dx from 0 to infinity

Homework Equations


Improper integrals

The Attempt at a Solution


Lim[tex]_{t->\infty}[/tex] [tex]\int[/tex][tex]\frac{t}{0}[/tex] ([tex]\frac{x}{(x^2+2)(x^2+2)}[/tex])

I tried integrating this by parts and also by partial fractions but neither seemed to lend itself nicely to the problem. (Choosing dv = (x^2+2)^(-2) made finding v ugly and based on the rules for choosing u shouldn't I choose x to be u?) And partial fractions didn't seem to work either. Any suggestions?
 
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  • #2
That's a pretty ugly tex post but if you mean the integral of x*dx/(x^2+2) try u=x^2+2.
 
  • #3
I am still trying to play with the formatting, sorry, I will write it out in words in the mean time: the integral of x over (x^2+2)^2 dx.

But, yes, it seems like that simple u-substitution will work! Thank you ... I feel so silly for overcomplicating the problem!
 

What is an improper integral?

An improper integral is an integral that does not have a finite value because one or both of the limits of integration are infinite or the integrand is not defined at some point within the interval of integration.

What is the difference between a Type 1 and Type 2 improper integral?

A Type 1 improper integral has one or both limits of integration that are infinite, while a Type 2 improper integral has an integrand that is not defined at some point within the interval of integration.

How do you evaluate an improper integral?

To evaluate an improper integral, you must first determine whether it is a Type 1 or Type 2 integral. Then, use the appropriate method (limit comparison, comparison, or the p-test) to determine if the integral converges or diverges. If it converges, use the limit of integration to evaluate the integral. If it diverges, the integral has no numerical value.

Can an improper integral have a finite value?

Yes, an improper integral can have a finite value if it converges. This means that the integral has a numerical value that can be calculated using integration techniques.

Are there any real-world applications of improper integrals?

Yes, improper integrals have many applications in physics and engineering, such as calculating the area under a curve, finding the center of mass of an object, and determining the amount of work done by a varying force. They are also used in probability and statistics to calculate probabilities and expected values.

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