baldymcgee6
  • baldymcgee6
How to setup this integral with partial fractions..?
Mathematics
chestercat
  • chestercat
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baldymcgee6
  • baldymcgee6
|dw:1358744328225:dw|
baldymcgee6
  • baldymcgee6
x^7
baldymcgee6
  • baldymcgee6
just have to set it up, don't have to actually integrate it. :)

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hartnn
  • hartnn
x(x^6-1) =x(x^3+1)(x^3-1)
baldymcgee6
  • baldymcgee6
\[\LARGE{\int\limits \frac{ 1 }{ x^7 - x }dx}\]
kirbykirby
  • kirbykirby
|dw:1358744642346:dw|
hartnn
  • hartnn
do you know general formulas for a^3+b^3 and a^3-b^3..
baldymcgee6
  • baldymcgee6
yes
baldymcgee6
  • baldymcgee6
so how would this spit into partials?
hartnn
  • hartnn
if there is quadratic equation in denominator, which cannot be factorized, then in numerator, it'll be linear expression, Ax+B...
kirbykirby
  • kirbykirby
For x^3-1 and x^3+1 you can use Long polynomial division by finding a factor (x-a) where a is a root of the polynomial
hartnn
  • hartnn
so, whats x^3+1=... ? x^3-1=.... ?
kirbykirby
  • kirbykirby
I never learnt the general formula for a^3+b^3. I think it's easier to do polynomial division because it will work for higher powers
baldymcgee6
  • baldymcgee6
x^3 +1 = (x+1)(x^2-x+1)
hartnn
  • hartnn
so, above x^2-x+1 will come a linear expression like Ax+B and above linear terms like x, x(x+1), constants like C,D,E ... will come.
baldymcgee6
  • baldymcgee6
@kirbykirby polynomial division of what and what?
baldymcgee6
  • baldymcgee6
x^3 -1 = (x-1)(x^2+x+1)
baldymcgee6
  • baldymcgee6
okay, i'll try this
hartnn
  • hartnn
so, you have 5 factors, out of which 2 is quadratic.
kirbykirby
  • kirbykirby
for x^3+1, you find the factors of the constant (here, its 1) and find which one = 0 when you plug it into your x-terms. So, -1 works: ((-1)^3+1) = 0. So, the factor that divides the polynomial is always of the form (x-a). So, (x-(-1)) = (x+1). Then divide x^3+1 into x+1
baldymcgee6
  • baldymcgee6
|dw:1358745155474:dw|
hartnn
  • hartnn
\(\checkmark \)
baldymcgee6
  • baldymcgee6
@kirbykirby that is useful to know rather than memorizing these formulas.
baldymcgee6
  • baldymcgee6
thanks for the help guys!
kirbykirby
  • kirbykirby
No problem :)
kirbykirby
  • kirbykirby
It's often called the "Factor Theorem". http://en.wikipedia.org/wiki/Factor_theorem

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