Is x*tan(x) integrable in elementary functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:21:46Z http://mathoverflow.net/feeds/question/108598 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108598/is-xtanx-integrable-in-elementary-functions Is x*tan(x) integrable in elementary functions? Victor 2012-10-02T04:36:40Z 2012-10-02T12:26:49Z <p>Dear All,</p> <p>I'm teaching Calculus and my students asked me to calculate the integral of x*tan(x).</p> <p>I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in elementary functions.</p> <p>Does anybody know how to calculate it, or otherwise prove it is not integrable in elementary functions?</p> http://mathoverflow.net/questions/108598/is-xtanx-integrable-in-elementary-functions/108606#108606 Answer by Benjamin Dickman for Is x*tan(x) integrable in elementary functions? Benjamin Dickman 2012-10-02T06:48:12Z 2012-10-02T06:48:12Z <p><strong>Some thoughts on this antiderivative:</strong></p> <p>Attacking $\log(\cos x)$ using integration by parts, we find:</p> <p>$$\int \log(\cos x) = x\log(\cos x) + \int x \tan x dx$$</p> <p>So the question has now become: how do we find an antiderivative for log(cos x)?</p> <p>Next, we observe that </p> <p>$$\cos x = \frac{1}{2}(e^{ix} + e^{-ix}) = \frac{1}{2}e^{ix}(1 + e^{-2ix})$$</p> <p>Taking the log of this, we end up with:</p> <p>$$-\log 2 + ix + \log(1 + e^{-2ix})$$</p> <p>Recall that we can write </p> <p>$$\log(1 + y) = \sum_{k = 1}^{\infty} \frac{(-1)^{k+1}y^{k}}{k}$$</p> <p>We can now apply this with $y = e^{-2ix}$ as above and integrate term by term.</p> <p>Putting all these pieces together will give you a (nasty) way to integrate $x\tan x$.</p> <p>As far as showing it's not integrable in elementary functions, I suspect your best bet would be an appeal to a theorem of Liouville. See, for example, <a href="http://math.hunter.cuny.edu/ksda/papers/churchill.pdf" rel="nofollow">this link</a>. (Sorry I can't be of more help here!)</p> <p>All that said, perhaps you could ask your students some form of the following: show</p> <p>$$\int x\tan^{2}x dx = x\tan x - \frac{x^2}{2} + \log(\cos x) + C$$</p> <p>(You can find this latter, more tractable problem and its solution written out in nice detail <a href="http://mathforum.org/library/drmath/view/53700.html" rel="nofollow">here</a>.)</p> http://mathoverflow.net/questions/108598/is-xtanx-integrable-in-elementary-functions/108616#108616 Answer by Peter Mueller for Is x*tan(x) integrable in elementary functions? Peter Mueller 2012-10-02T09:48:12Z 2012-10-02T12:26:49Z <p>Finding an anti-derivative of $x\tan x$ amounts to finding an anti-derivative of $f=\frac{x}{e^x+1}$. Consider the field $K=\mathbb C(x,e^x)$. Note that $K$ is closed under taking derivatives. If $f$ is elementary integrable, then Liouville's Theorem gives elements $u_i\in K$, $\gamma_i\in\mathbb C$, $v\in K$ with $$\frac{x}{e^x+1}=\sum\gamma_i\frac{u_i'}{u_i}+v'.$$ Consider the $u_i$ and $v$ as rational functions in $e^x$ with coeffcients in $\mathbb C(x)$. By the property of the logarithmic derivative we may assume that the $u_i$ are actually distinct irreducible monic polynomials with respect to $e^x$, or elements from $\mathbb C(x)$.</p> <p>Looking at poles (with respect to the `variable' $e^x$) shows that at most one of the $u_i$ is $e^x+1$, and the other $u_i$'s are in $\mathbb C(x)$. Similarly, we see that $v\in\mathbb C(x)$. So there indeed must be one index $i$ with $u_i=e^x+1$. However, $\frac{x}{e^x+1}-\gamma_i\frac{u_i'}{u_i}=\frac{x}{e^x+1}-\gamma_i\frac{e^x}{e^x+1}$ isn't in $\mathbb C(x)$, a contradiction.</p> <p><em>Remark:</em> The argument given here is somewhat sketchy, some routine details need to be filled in, like that $u_i'$ and $u_i$, as polynomials in $e^x$, are relatively prime. A beautiful paper about Liouville's Theorem is Rosenlicht's article <a href="http://www.jstor.org/stable/2318066?origin=crossref" rel="nofollow">Integration in finite terms</a>. My argument somewhat follows Rosenlicht's example of finding an anti-derivative of $f(x)e^{g(x)}$, where $f$ and $g$ are rational functions.</p>