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December 17th, 2017, 08:53 PM   #11
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Yes, you can put me on your ignore list. Bai.
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December 26th, 2017, 08:05 AM   #12
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Leibniz's Rule

Here is a propose solution.
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December 26th, 2017, 10:19 AM   #13
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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.)
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Last edited by Country Boy; December 26th, 2017 at 10:25 AM.
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December 26th, 2017, 12:52 PM   #14
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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.
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