Integrating Partial Fraction: Steps & Examples

[LusTRouZ]

I need to integrate the function 1/(X^7 + x) which simplifies to 1/x(x³ + 1)(x³ - 1) or any other problem where the degree of the denominator is at least 2 larger than the numerator. how do I do this?

 

[Goongyae]

>1/(X^7 + x) which simplifies to 1/x(x3 + 1)(x3 - 1)

No, it doesn't.

>integrate the function 1/(X^7 + x)

First solution:
Note that 1/(X^7 + x) = 1/x * 1/(1+x^6)
= 1/x * (1 -x^6 + x^12 - x^18 + ...)
=1/x - x^5 + x^11 - x^17 + ...

Then the integral is
C + log(x) - x^6/6 + x^12/12 - x^18/18 + ...
the series part is clearly a logarithm expansion, so it's
C + logx - log(1+x^6)/6

Second solution
1/(X^7 + x) = x^-7 * 1/(1+x^-6)
= x^-7 * (1 - x^-6 + x^-12 + x^-18 + ...)
=x^-7 - x^-13 + x^-19 - ...
Then the integral is
C -x^-6/6 + x^-12/12 - x^-18/18 + ...
i.e. it's
C - log(1+x^-6)/6
which is equal to the above solution.

Third solution
1/(X^7 + x) = x^-7/(1+x^-6)
and the solution
is C - log (1+x^-6)/6 based on inspection (it clearly produces the correct derivative).

 

[Mark44]

I need to integrate the function 1/(X^7 + x) which simplifies to 1/x(x³ + 1)(x³ - 1) or any other problem where the degree of the denominator is at least 2 larger than the numerator. how do I do this?

Your factorization is incorrect. x⁷ + x = x(x⁶ + 1) $\neq$ x(x³ + 1)(x³ - 1).

$$\frac{1}{x^7 + x} = \frac{1}{x(x^6 + 1)} = \frac{1}{x(x^2 + 1)(x^4 - x^2 + 1)}$$

You can break up that quartic into x² - (1/2 + sqrt(3)/2) and x² - (1/2 - sqrt(3)/2).

 

[LusTRouZ]

is there any way of doing this with partial fractions?

 

[Goongyae]

Yes, of course there is a partial fraction solution.

Fourth solution
1/(x+x^7) = 1/x * 1/(1+x^6) = 1/x * 1/((1+ax)*(1+bx)*(1+cx)*(1+dx)*(1+ex)*(1+fx))

where a,b,c,d,e,f are the six roots of unity exp(2 i pi N/6) for N = 0...5

= 1/(6x) * (1/(1+ax) + 1/(1+bx) + ... + 1/(1+fx))

This integrates to
C - (1/6)[ log(a+1/x) + log(b + 1/x) + log(c+1/x) + ... + log(f+1/x)]

which is the same as three other solutions listed above.