Solve the integral: \int\frac{1}{x^{2n}+1}dx

Kelly Nelson 2022-01-03 Answered
Solve the integral:
1x2n+1dx
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Elaine Verrett
Answered 2022-01-04 Author has 41 answers
We have f(x)=1xn+1. Note that we can write
f(x)=k=1n(xxk)1 (1)
where xk=ei(2k1)πn k=1,,n
We can also express (1) as
f(x)=k=1nak(xxk)1 (2)
where ak=xkn
Now, we can write
1xn+1dx=1nk=1nxklog(xxk)+C
which can be more explicitly written as
1xn+1dx=1n
k=1n(12xkrlog(x22xkrx+1)xkiarctan(xxkrxki))+C
where xkr and xki are the real and imaginary parts of xk respectively, and are given by
xr=Re(xk)=cos((2k1)πn)
xki=Im(xk)=sin((2k1)πn)
Note 1:
The integral of 11+x2n is a special case for the development herein. Simply let n2n
Note 2:
As requested, we will derive the form ak=xkn. To that end, we use (2) and observe that
limxxl(
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Tiefdruckot
Answered 2022-01-05 Author has 46 answers
It has no simple closed form, unless you also give some nice integration endpoints, such as 0. For your curiosity, you get x2F1(12n,1,1+12n,x2n), where 2F1 is the hypergeometric function.
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karton
Answered 2022-01-11 Author has 439 answers

First find x2n+1=0, then split into sum of fractions 1/(x+b) and 1/(x+b) and 1/(x2+bx+c) and integrate those. I seem to have forgotten what it's called in English. Not partial integration or integration by parts. Partial fraction decomposition maybe?

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