Calc II - Integration of Partial Fractions

[demersal]

Homework Statement

Hi everyone, here is a new partial fractions question I just cannot understand:

\int\frac{x^{3}}{x^{3}+1}dx

Homework Equations

Partial Fractions, difference of perfect cubes, polynomial long division

The Attempt at a Solution

\int\frac{x^{3}}{x^{3}+1} dx

\int1 dx + \int\frac{-1}{x^{3}+1} dx

x + \int\frac{-1}{(x+1)(x^{2}-x+1)} dx

\frac{A}{x+1} + \frac{B}{(x^{2}-x+1)}

A(x^{2}-x+1) + B(x+1) = - 1

If I use coefficients:
x^{2}: 0=A
x: 0 = B-A
: -1 = A+B

These don't add up! Also, using critical values leads me to a similar problem. What am I doing wrong in my strategy of attacking these partial fractions??

Thank you for your time and help, you all are so wonderful!

 

[Pacopag]

\frac{A}{x+1}+\frac{B}{(x^{2}-x+1)}

Remember that when you decompose the fraction, the numerator must be a general polynomial with degree one less than that of the denominator. What I'm getting at is that the numerator in your "B"-term is wrong. What should it be instead?

 

[demersal]

Hmm, I don't quite understand what you are saying. I thought I was just supposed to multiply the numerator by the denominator and cancel. Is there another concept I am missing?

 

[statdad]

If the denominator is linear, the numerator must be a constant. If the denominator is an irreducible quadratic, the numerator should be a ********** (hint: not a constant)

 

[gabbagabbahey]

The numerator of each term must be a polynomial of degree one less than the degree of the denominator. x^2-x+1 is a 2nd degree polynomial, so the numerator for that term should be a first degree polynomial; Bx+C.

 

[Pacopag]

In your second term, you just put a B in the numerator. But that is a zeroth order polynomial when you have a second order polynomial in the denominator. So instead of just B, you need to put in an arbitrary first order polynomial. What is another name for a first order polynomial?

 

[Pacopag]

Right gabbagabbahey. Notice that the largest exponent in the denominator is 2. So the largest exponent in the numerator should be 1 (i.e. one less than 2).

 

[demersal]

So it should be Bx?

 

[demersal]

ohh I see gabbagabbahey, I should have a Bx and a separate C over the same denominator?

 

[gabbagabbahey]

ohh I see gabbagabbahey, I should have a Bx and a separate C over the same denominator?

You should have:
\frac{A}{x+1}+\frac{Bx+C}{(x^{2}-x+1)}

 

[Pacopag]

Yes. By "general" or "arbitrary" we mean that you have to include all the terms. e.g. for a second order polynomial you'd have
Ax^2+Bx+C.
You must include all the lower order terms too (i.e. the 2nd (A), 1st (B) and 0th (C) terms).

 

[demersal]

Ok! I got it now! Thank you both! Hopefully I'll never have to post about partial fractions again :)