## NotSObright Group Title integral((x^3+x+1)/(x^4+x^2+1)) 2 years ago 2 years ago

1. DHASHNI Group Title
2. NotSObright Group Title

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

3. rivermaker Group Title

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

4. rivermaker Group Title

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

5. rivermaker Group Title

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