Integrate using partial decomposition indefinite integral (1/(x^(4)+1)dx. P.S. 4 is the only exponent.
Thanks
First, find the poles (i.e. singularities) of the function. It is easy to see that they are at:
x1 = exp(i pi/4)
x2 = exp(i 3/4 pi)
and the complex conjugates of these to find the two other poles:
x3 = x1-bar = exp(-i pi/4)
x4 = x2-bar = exp(-i 3/4 pi)
You then expand the function around each of the poles, keeping only the singular terms and add up the results:
1/(4 x1^3) 1/(x - x1) +
1/(4 x2^3) 1/(x - x2) + complex conjugate (for real x)
is equal to 1/(x^4 +1)
This follows from the fact that
1/(x^4 +1) minus the four terms has no singulariteis, so it is a omplex analytical function. This function tends to zero at infinity, so it is boiunded. From Liouville's theorem:
http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)
it then follows that this function must be zero everywhere.
So, we can write the integral as:
1/(4 x1^3) Log(x-x1) +
1/(4 x2^3) Log(x-x2) + complex conjugate =
1/2 Re[exp(-3 i pi/4)
Log(x - exp(i pi/4))] +
1/2 Re[exp(-9 i pi/4)
Log(x - exp(3 i pi/4))]
You can rewrite this by writing the arguments of the logs in polar form and using that:
Log(r exp(i theta)) = Log(r) + i theta
Since theta involves the arctan function, you then get the expression in conventional form, but then with much less work than using the usual (high school) method of partial fractions.