Indefinite Improper Integrals

[NaN-Gram]

Hello everyone, I wanted to get some more help with integrals.

Specifically, I had a question about the following integral.



I had also done some work relating to the expression as well, listed below.


I had done some work leading up to the final expression, but I got stuck.
I was wanting to express the integral as either ((x squared) over 2) times arctangent squared, or ((x squared) over 2) plus arctangent squared.
What's the right step forward in this case?

 

[Steven G]

If you let u=x^2 + 1, then du = 2xdx is right there! If you are going to make a u-substitution then you must do it carefully. You must write each and every part (did I mention every part?) of the integral in terms of u. Give that a try and post back.

 

[NaN-Gram]

I didn't think of that, I'll be back soon.

It's fair to substitute u for x, as I assume that at the very least, infinity plus one is still infinity.

 

[NaN-Gram]

Here's some new work I've done on the problem.



I think that it might have the answer, since the limits of infinity are removed.

I made a couple of typos, but I think that it all works out.

 

[Steven G]

I didn't think of that, I'll be back soon.

It's fair to substitute u for x, as I assume that at the very least, infinity plus one is still infinity.

I do not think that you meant what you said. If you replace x with u, then dx gets replaced with du and all that changes is the x's become u's. It is still the same exact integral.
Now maybe you meant to replace x+1 or x^2 +1 with u is ok since x or u is going to infinity. Maybe that works for limits but this is an integral where we are adding up areas from say -5 to 7 (along with -infinity to -5 and 7 to infinity) and replacing x+1 or x62 + 1 with u is not correct.

 

[pka]

Hello everyone, I wanted to get some more help with integrals.
Specifically, I had a question about the following integral.
View attachment 17330I had also done some work relating to the expression as well, listed below.

You might note that \(f(x)=\dfrac{2x}{(x^2+1)^2}\) is an odd function, \(f(-x)=-f(x)\), as such \(\displaystyle\int_{ - c}^c {f(x)dx} = 0,\;c > 0\)

 

[Steven G]

Prof, I was just about to post that since the integral converged then the answer had to be 0 since the integrand is odd. Even though an integrand is odd, it could still diverge. For example if f(x) = x then the integral would have been divergent. Is there any test that would tell whether an integral of an odd function with limits at +/- infinity would be convergent (to 0)?

 

[pka]

For example if f(x) = x then the integral would have been divergent. Is there any test that would tell whether an integral of an odd function with limits at +/- infinity would be convergent (to 0)?

You are correct. I was in quite a hurry but should have said more. Chapter 7 of Smith & Minton has a good discussion.