This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, Hostetler, and Edwards". This problem is at the end of the first section of the chapter 8 exercises. Here's the problem:

Evaluate $$\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$$

Please. Any help is very much appreciated. So are solutions. Thank you!

Edit: I like the solution given, but I was interested to see if there is any other way of doing the problem? I'm excited to see the results.

  • 1
    $\begingroup$ In general, when there's a symmetry (such as around $x=3$ in this problem), it's a good idea to try to exploit that symmetry. Which is more easily said once you've seen the answer. $\endgroup$ – Teepeemm Oct 4 '14 at 19:08
  • $\begingroup$ Yes. I had a feeling that there was an inherent symmetry, but I was doubting myself so I didn't take a serious enough look at it. Glory is achieved by the fearless and not the second guessers. I need to start taking symmetry much more seriously! Thank you for the advice. $\endgroup$ – Saudman97 Oct 4 '14 at 20:26

Let $$ \mathcal{I}=\int_{2}^{4}\dfrac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)}+\sqrt{\ln(3+x)}}\,\mathrm{d}x $$ Now, use that $$ \int_{a}^{b}f(x)\,\mathrm{d}x\overset{(1)}{=}\int_{a}^{b}f(a+b-x)\,\mathrm{d}x $$ Then, $$ \mathcal{I}=\int_{2}^{4}\dfrac{\sqrt{\ln(3+x)}}{\sqrt{\ln(3+x)}+\sqrt{\ln(9-x)}}\,\mathrm{d}x $$ Add up these two integrals to get $$ 2\mathcal{I}=\int_{2}^{4}\dfrac{\sqrt{\ln(9-x)}+\sqrt{\ln(3+x)}}{\sqrt{\ln(9-x)}+\sqrt{\ln(3+x)}}\,\mathrm{d}x $$ Thus, $$ \mathcal{I}=1 $$

In order to prove $(1)$, write the integral using another variable, say, $t$: $$ \int_{a}^{b}f(a+b-x)\,\mathrm{d}x=\int_{a}^{b}f(a+b-t)\,\mathrm{d}t $$ In the latter one, set $x=a+b-t$ and $\mathrm{d}t=-\mathrm{d}x$ and change the limits of integration to obtain $$ \begin{aligned} \int_{a}^{b}f(a+b-t)\,\mathrm{d}t&=-\int_{b}^{a}f(x)\,\mathrm{d}x\\ &=\int_{a}^{b}f(x)\,\mathrm{d}x. \end{aligned} $$

  • 2
    $\begingroup$ There is an interesting (very) related article published in MATHEMATICS MAGAZINE VOL. 81, NO. 2, APRIL 2008, and entitled ''Lazy Student Integrals'' By GREGORY GALPERIN & GREGORY RONSSE. $\endgroup$ – Idris Dec 14 '14 at 5:10
  • $\begingroup$ Very Nice Solution. $\endgroup$ – juantheron Aug 12 '15 at 15:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.