# Integrating a Rational Function

I am studying for a test and I am trying to evauate the integral below. I know how to simplify it with partial fractions, but when I try to solve it, I cannot seem to find a substitution that will simplify it enough to solve in reasonably quick . I plugged it into wolfram and as usual it doesn't give a quick way either. If anybody has a way this integral can be solved quickly, as there will be a time crunch on my test. $$\int\frac{x^3+x+2}{x^4+2x^2+1}dx$$

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From partial fractions and two variable substitutions \begin{align} \int \frac{x^3+x+2}{x^4+2x^2+1}dx &= \int\frac{x}{x^2+1}dx + 2\int\frac{1}{x^2+1}\frac{1}{x^2+1}dx \\ &= \frac{1}{2}\int\frac{du}{u}+\int\frac{1}{(\tan v)^2+1}dv \end{align}

where $u=x^2+1$ and $v=\arctan x$ (recall that $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$). Now since $\tan^2v+1 = \sec^2 v = 1/\cos^2 v$,

\begin{align} \int \frac{x^3+x+2}{x^4+2x^2+1}dx &= \frac{\ln u}{2} + 2\int \cos^2(v)\, dv \\ &= \frac{\ln u}{2}+2\left(\frac{\sin v \cos v}{2} +\frac{v}{2}\right) \\ &= \frac{\ln \left(x^2+1\right)}{2}+\sin \left(\arctan x\right) \cos \left(\arctan x\right) +\arctan x \\ &= \frac{1}{2}\ln \left(x^2+1\right) + \frac{x}{x^2+1} + \arctan x \end{align} Hope this helps.

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$\int \frac{x(x^2+1)+2}{(x^2+1)^2} dx$
The integral splits into $\int \frac{x}{x^2+1} dx + \int \frac{2}{(x^2+1)^2} dx$. Which integral are you having problems with? –  fixedp Feb 24 at 2:38
If you know about integration by parts, that would be easier. If not, $x=\tan(u)$ will give you the correct answer after a bit of work. Both methods can be found here: math.stackexchange.com/questions/35924/integral-of-frac11x22 –  fixedp Feb 24 at 2:47