Maximizing Integration Efficiency: Long Division vs Partial Fractions

[Cacophony]

Homework Statement

∫(2x+1)/(x²+2x+1)(x²+x+1)

Homework Equations

none

The Attempt at a Solution

I've foiled this out to look like:

∫(2x+1)/(x^4+3x³+4x²+3x+1)

I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or should I use a different approach?

 

[SteamKing]

I would try a different approach.

Look at the denominator and see if it can be factored rather than multiplied together.

 

[eumyang]

I've foiled this out to look like:

∫(2x+1)/(x^4+3x³+4x²+3x+1)

I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or should I use a different approach?

Why would you try long division? You are dividing something "smaller" into something "bigger". It is as if you are suggesting that to evaluate $\frac{7}{584}$ you divide 584 by 7.

Have you considered using partial fractions to split the integral into two?

 

[iRaid]

Definitely do partial fractions. The only time you do long division is when the degree on top is bigger than the bottom.

 

[Mark44]

Definitely do partial fractions. The only time you do long division is when the degree on top is bigger than the bottom.

Or when the degrees are equal.