Calculus Calculus Math Forum

 December 17th, 2017, 08:53 PM #11 Senior Member   Joined: Sep 2015 From: USA Posts: 2,638 Thanks: 1474 Yes, you can put me on your ignore list. Bai. December 26th, 2017, 08:05 AM   #12
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Leibniz's Rule

Here is a propose solution.
Attached Images Problem_2.jpg (86.5 KB, 4 views) December 26th, 2017, 10:19 AM #13 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 It's easy to check, by actually doing the integral and then differentiating it. $F(x)= \int_{2x}^{x^2}\frac{2t}{t^2+ 1}dt$ Let $u= t^2+ 1$. Then $du= 2tdt$, When $t= 2x$, $u= 4x^2+ 1$ and when $t= x^2$, $u= x^4+ 1$, so the integral becomes $F(x)= \int_{4x^2+ 1}^{x^4+ 1} \frac{du}{u}= \left[ ln(u)\right]_{4x^2+ 1}^{x^4+ 1}= ln\left(\frac{x^4+ 1}{4x^2+ 1}\right)$ The derivative of that is $F'(x)= \frac{4x^2+ 1}{x^4+ 1}\frac{4x^3(4x^2+ 1)- 8x(x^4+ 1)}{(4x^2+ 1)^2}$. $F'(2)= \frac{17}{17}\frac{32(17)- 16(17)}{17^2}= \frac{16}{17}$ Well done! (I strongly suggest you do NOT block romsek. He thought, from you first post, that you were asking us to do a "take home" test for you and, after realizing that you were practicing for a test, apologized. If you block romsek you will only prevent yourself from getting help from one of the best here.) Thanks from romsek and JeffM1 Last edited by Country Boy; December 26th, 2017 at 10:25 AM. December 26th, 2017, 12:52 PM #14 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond The given integral has an easy antiderivative (ignoring the constant of integration). Compute that and then some manipulation according to log identities gets you an answer. Not mentally demanding at all, if you know as much as you should, but perhaps somewhat tedious. Tags limit, question 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 jamesmith134 Calculus 16 November 9th, 2014 04:46 AM Tzad Calculus 4 February 10th, 2014 05:45 PM wannabe1 Real Analysis 2 September 30th, 2010 01:39 PM Botnaim Calculus 2 April 4th, 2008 12:31 PM Tzad Algebra 2 December 31st, 1969 04:00 PM

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