Efficient Integration Tips: Simplifying t^2/(1+t^4) with Substitution Method

[Oster]

I want to integrate the function t^2/(1+t^4)

I tried substituting t^2 as tan(x) but that requires me to integrate the root of tan(x).

Any suggestions?

 

[GreenPrint]

Oster

assuming your really can't evaluate using u substitution as a first step, or other simpler ways of evaluating, and if your work is indeed right... I would then move onto asking you...

Do you know the "cases" of this type of simple rational function that help you determine when to long divide or use partial fractions?

If your just taking the integral of a simple rational function

P(x)/Q(x)

there are certain things that you can look for to determine when to long divide or use partial fractions... assuming you can even take the integral... which you can in this case... I would try looking into using one of these methods sense u substitution doesn't seem to work for you... or at least I guess you don't know how to proceed using u substitution... maybe you can't solve the integral using u substitution as a first step if you don't know how to proceed... at least that's what I would assume if I were in your shoes... unless your doing something terribly wrong... but if your not making a mistake look into long dividing and using partial fractions before looking into other methods of evaluating it... do as little work as possible... if you can't evaluate using simple rules move onto u substitution, the next simplest way of evaluating in most cases, as a first step then move onto the next simplest thing if that doesn't work, long dividing or partial fractions in most cases... before considering other methods on how to further proceed...

GreenPrint

 

[SammyS]

As GreenPrint suggests, you may want to try partial fractions.

1+t⁴ can be factored.

$1+t^4 = (t^2 + (\sqrt{2}) t + 1 )(t^2 - (\sqrt{2}) t + 1 )$

 

[Oster]

Oooo. I could never have figured out that factorization on my own. Thank you.

 

[hunt_mat]

The way to see this is to write:
$$1+t^{4}=1+t^{4}-2t^{2}+2t^{2}=(1+t^{2})^{2}-2t^{2}=(1+\sqrt{2}t+t^{2})(1-\sqrt{2}t+t^{2})$$
I am rather jealous that SammyS, got the answer before I remembered the trick to solving this one.

 

[Bohrok]

I want to integrate the function t^2/(1+t^4)

I tried substituting t^2 as tan(x) but that requires me to integrate the root of tan(x).

Any suggestions?

Actually, integrating that expression always comes up when integrating $\sqrt{\tan x}$!

Another way of integrating it without using partial fractions involves completing the square in the denominator
https://www.physicsforums.com/showpost.php?p=792629&postcount=12

 

[SammyS]

@ hunt_mat: Don't be too jealous. I used WolframAlpha to get the factors. Thanks for showing the 'trick'! -- basically completing the square (in a weird way) to get the difference of squares.

@ Bohrok: Thanks for that link!

 

[Dick]

Oooo. I could never have figured out that factorization on my own. Thank you.

Just for the record, any real polynomial can be factored into real linear and quadratic factors. You can find them if you can find the roots of the real polynomial. Real roots lead to real linear factors, obviously. But if you have a complex root c, then c* is also a root, so (t-c)(t-c*) is your real linear factor. It's easy to find the complex roots of t^4+1 using deMoivre.

 

[hunt_mat]

Just for the record, any real polynomial can be factored into real linear and quadratic factors. You can find them if you can find the roots of the real polynomial. Real roots lead to real linear factors, obviously. But if you have a complex root c, then c* is also a root, so (t-c)(t-c*) is your real linear factor. It's easy to find the complex roots of t^4+1 using deMoivre.

Of course, something that we should have all known!