- #1
demersal
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Homework Statement
Hi everyone, here is a new partial fractions question I just cannot understand:
[tex]\int[/tex][tex]\frac{x^{3}}{x^{3}+1}[/tex]dx
Homework Equations
Partial Fractions, difference of perfect cubes, polynomial long division
The Attempt at a Solution
[tex]\int[/tex][tex]\frac{x^{3}}{x^{3}+1}[/tex] dx
[tex]\int[/tex]1 dx + [tex]\int[/tex][tex]\frac{-1}{x^{3}+1}[/tex] dx
x + [tex]\int[/tex][tex]\frac{-1}{(x+1)(x^{2}-x+1)}[/tex] dx
[tex]\frac{A}{x+1}[/tex] + [tex]\frac{B}{(x^{2}-x+1)}[/tex]
A(x[tex]^{2}[/tex]-x+1) + B(x+1) = - 1
If I use coefficients:
x[tex]^{2}[/tex]: 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!