How Do You Integrate w/(w^4 + 1)?

[chubb rock]

I'm supposed to integrate w/w^4 + 1

The two ideas I'd come up with was trying to substitute w^4+1 with u but that results in a 0 in the numerator.

Then I thought maybe I have to put w/w^4+1 into the 1/1+w^2 form and integrate into 1/k tan^-1kx + C but I'm not sure how to keep the original function while making the w^4 a w^2.

Help please? Am I at least going in the right direction with any of these ideas?

 

[dx]

Use integration by parts. You'll have to do it twice.

EDIT: Mark44's way is better, ignore this.

 

[Mark44]

You're going in the right direction thinking of tan^-1(x).

An ordinary substitution should do the trick: u = x², so du = 2xdx. Then your integrand becomes roughly du/(u² + 1). Notice the parentheses I added to make clear what the denominator is. You'll also need to add the right constant multiplier in the numerator, since I omitted it.

 

[chubb rock]

Does that mean I can substitute just the variable and leave the "+ 1"? That is substitute u as w^2 getting u^2 = u^4?

 

[dx]

You mean u² = w⁴.

 

[Mark44]

Yes. The substitution is u = w², so u² = w⁴, and du = 2wdw.

 

[chubb rock]

Oh, yeah. Typo. That's what I meant.

Awesome, thanks for the help.