Question

Can someone please show and explain step by step how to evaluate this integral?

Problem #1)∫ 1/(x^3+x^2+x+1) dx


This is what I got so far. I am not sure what to do thereafter after doing partial fraction decomposition (in which I got A = 1/2, B = 1/2, and C = 1/2)

1/(x^2+1)(x+1) = ∫(-1/2x)+(1/2)/(x^2+1) + ((1/2)/(x+1)) dx

Answer 1

First step:
decompose into partial fractions (probably the hardest part of the problem)
Factorize the denominator (if possible)
x^3+x^2+x+1=x^2(x+1)+1(x+1)=(x^2+1)(x+1)
so we can decompose the given fraction into:
1/(x^3+x^2+x+1)
=1/2(x+1) + (1-x)/2(x^2+1)
which can be integrated readily.