Need some integration help!

[Sukhoi]

I have a bunch of integrals to do. They're quite complicated, but I think there is probably one procedure that will solve most of them. Here are some examples:

• 1/(x^6-1)
• x/(x^2+4x+3)
• 4/(x^3+4x)
• 1/(5x^2+8x+5)

Any idea what to do? U-substitution won't work. I don't think integration by parts will work.. Thanks

 

[Random Variable]

EDIT: I finally figured it out. You have to use partial fractions. God, I feel stupid.

 

[Shalmanese]

yup, good old Partial Fractions. IF you got Maple or something on your computer then it can do the factorising for you *hint* *hint*

 

[Sukhoi]

Ah, I forgot about partial fractions. We just got a quick lesson on them last week.

I think the third one will be pretty easy to do since the x will factor out of the bottom. Then I can make the two fractions real easy.

But what about the first one for example? If I tried to make a second fraction wouldn't it just be 1/1?

Edit: I have a TI-89, so I can get the answer. But I need to show all my work for the integration.

 

[Sukhoi]

My teacher didn't explain very well how to break it up into the separate fractions.

Would the third one be (4/x) - (4/(x^2+4)) ? I'm not sure if I put the 4 in the numberator of both or not. Thanks!

 

[Random Variable]

<< But what about the first one for example? >>


The first one is very nasty. X^6+1 factors into (x-1)(x+1)(x^2+x+1)(x^2-x+1).

The second one is more reasonable.

 

[Random Variable]

<< Would the third one be (4/x) - (4/(x^2+4)) ? I'm not sure if I put the 4 in the numberator of both or not. Thanks! >>


I would use trigonometric substution for the third one.

 

[j0lly]

Check your answer using the Integrator.

 

[Shalmanese]

First one is messy but still manageble. Just hope youve learnt all about solving simultaneous eqns using matrices.

third one is wrong but you got the factors right

To check if it is right or not, just expand it back out.

Vespian: how would you use trig substitution?

 

[Random Variable]

<< Vespian: how would you use trig substitution? >>


well, you could let let x = 2*tan(A) so dx = 2*sec^2(A)dA and proceed from there

EDIT: On second second thought, stick to partial fractions. Trigonometric substitution just makes the problem harder.

 

[Random Variable]

<< Would the third one be (4/x) - (4/(x^2+4)) ? I'm not sure if I put the 4 in the numberator of both or not. >>



4/(x(x^2+4)) = A/x + (Bx+C)/(x^2+4)

4 = Ax^2 + 4A + Bx^2 +Cx

A + B = 0
C = 0
4A=4
A=1
B=-1

1/x - x/(x^2+4)

 

[Sukhoi]

Thanks for all the help guys! I've got some of them figured out.

 

[Mookow]

My advice... bend over, and dont forget the lube!!! . Integration sucks.