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 July 29th, 2014, 07:10 AM #1 Newbie   Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0 Integral Hi Everybody! I am trying to prove analytically that the integral $\int_0^\infty \frac{r^2}{r^3+0.8} \frac{0.2\sin(11r^3+1.8 )}{11r^3+1.8}dr$ will be nonnegative. Can somebody give me an idea how to do it please. Is it even possible? July 29th, 2014, 07:38 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra Try writing the integral as an infinite sum of integrals such that the integrand is alternately positive and negative for the whole of the range of the integral. I think that you should be able to prove that each negative integral has a smaller magnitude that the preceding positive integral. This would give your solution. Thanks from Rtep July 29th, 2014, 08:27 AM   #3
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 Originally Posted by v8archie Try writing the integral as an infinite sum of integrals such that the integrand is alternately positive and negative for the whole of the range of the integral. I think that you should be able to prove that each negative integral has a smaller magnitude that the preceding positive integral. This would give your solution.
I were thinking about that, but what to do with the first two areas?
Attached Images graph.JPG (9.5 KB, 4 views) July 29th, 2014, 06:30 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra They are very, very close in area, but I think the first is a fraction larger. You might want to prove it though! The first area. The second area. Thanks from Rtep July 31st, 2014, 07:53 AM #5 Newbie   Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0 Its almost obvious that magnitude will decrease and that the distance between zeroes will become smaller and smaller. Thus, assuming that shape will remain the same, it follows that $\left|\int_{r_i}^{r_{i+1}}Integrand \ \ \ dr \right|\ge \left| \int_{r_{i+1}}^{r_{i+2}}Integrand \ \ \ dr\right|$, where $r_i$ are zeroes of the integrand. But what to do with the first two areas I have no idea.......I mean how to show analitically that the first area will be greater or equal to the second one? July 31st, 2014, 08:03 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra For the first four regions ($r \le 1$), you might have to settle for a numerical solution. But for $r \gt 1$, you ought to be able to get an analytic solution. July 31st, 2014, 08:05 AM #7 Newbie   Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0 Thank you. July 31st, 2014, 09:40 AM #8 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra Warning: that root isn't at 1, but $r^3 + 1.8 = 4\pi \implies r \gt 1$. Tags integral Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post gen_shao Calculus 2 July 31st, 2013 10:54 PM Obsessed_Math Calculus 4 February 9th, 2012 04:18 PM maximus101 Calculus 0 March 4th, 2011 02:31 AM xsw001 Real Analysis 1 October 29th, 2010 08:27 PM maximus101 Algebra 0 December 31st, 1969 04:00 PM

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