Integral of 1/(x^3 + 1)

Integral of 1/(x^3 + 1)
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[@mdasifeqbal2323]

Great method.

[@AntonioTiagoNunes-kf6ml]

Very beautiful.

[@renesperb]

A straightforward method , which works the same way for the general integrand 1/ (x^n+a) , is partial fraction decomposition .
You need the zeros xk of the denominator . In the present case they are -1,and 1/2(1±i √3). The partial fraction coefficients
are 1/(3*xk^2) = xk/3*xk^3= -xk/3. The integrals are then easy

[@grrgrrgrr0202]

I really wish you elaborated on how to integrate rational functions in general. The general method is to write the function as a sum of rational functions where the denominators are powers of irreducible linear or quadratic terms. Then the problem is essentially reduced to integrating the following functions:
* 1/(x^2+1)
* 1/(x^2+1)^n; n>1
* x/(x^2+1)^n
* 1/x

[@franciscojaviermartinez7976]

Tan^-1 = arctan.

[@alexchan4226]

x/2(x^3 +1）+c

[@51038HK]

1/3ln|x^3+1| error , 1/3ln|x+1| correct