Question

integral((x^3+x+1)/(x^4+x^2+1))

Answer 1

http://www.wolframalpha.com/input/?i=integral%28%28x^3%2Bx%2B1%29%2F%28x^4%2Bx^2%2B1%29%29

Answer 2

I am concerned with what substitution has to be carried out and not the answer

Answer 3

you have the form P(x)/Q(x) where P and Q are polynomials. Further the degree of the numerator, namely 3, is one less than the denominator. In such situations, find \[dQ/dx \] and express P in terms of that.
\[\int\limits{dy/y} = \log y\]
the remaining terms are hopefully simpler.
If you need more help ask

Answer 4

\[d/dx ({{x^4+x^2+1}}) = 4x^3 + 2x \]
so
\[P(x) = {1/4}({dQ/dx}) + (1/2)x + 1 \]
So the original problem reduces to integrating
\[dQ/Q + (1/2)\int\limits x dx/Q + \int\limits dx/Q \]

Answer 5

Sorry the last equation should read:
\[(1/4)\int\limits dQ/Q + (1/2)\int\limits\limits x dx/Q + \int\limits\limits dx/Q\]