 My Math Forum Please explain. (Trigonometric substitution)

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January 14th, 2016, 03:20 AM   #1
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I tried
$\displaystyle x^2 + 16 = 16sec^2\theta\\ 2x dx = d/dx (16sec^2\theta)$
and got stuck here
Attached Images Screenshot.jpg (4.2 KB, 14 views) January 14th, 2016, 05:09 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 What, exactly is the question? To integrate $\displaystyle \sqrt{x^2+ 16}$? Then you want to use the trig identity $\displaystyle tan^2(\theta)+ 1= sec^2(\theta)$ to "get rid of" that square root: if $\displaystyle x= 4 tan(\theta)$ then $\displaystyle \sqrt{x^2+ 16}= \sqrt{16tan^2(\theta)+ 16}= 4\sqrt{tan^2(\theta)+ 1}= 4\sqrt{sec^2(\theta)}= 4 sec(\theta)$. Differentiating both sides of $\displaystyle x= 4tan(\theta)$ (not $\displaystyle x^2+ 16= 4 sec^2(\theta))$ we have $\displaystyle dx= 4 sec^2(\theta) d\theta$ so $\displaystyle \int \sqrt{x^2+ 16} dx= 16 \int sec^3(\theta) d\theta$. To integrate that, I think I would write it as $\displaystyle 16\int \frac{d\theta}{cos^3(\theta)}$, then multiply both numerator and denominator by $\displaystyle cos(\theta)$ to get $\displaystyle 16\int \frac{cos(\theta)d\theta}{cos^4(\theta)}= 16\int \frac{cos(\theta)}{(1- sin^2(\theta))^2}$ and let $\displaystyle v= sin(\theta)$. Thanks from Volle January 14th, 2016, 05:14 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,835 Thanks: 2162 Let $x = 4\tan(θ)$, where $|θ| < \pi/2$, so that $dx = 4\sec^2(θ)dθ$. $\displaystyle \int\!\sqrt{x^2+16}\,dx = 16\!\int\!\sec^3(θ)dθ$ Integrating by parts, $\displaystyle \int\!\sec^3\!(θ) dθ = \sec(θ)\tan(θ) - \!\int\!\sec(θ)\tan^2(θ)dθ = \sec(θ)\tan(θ) - \!\int\!\sec(θ)(\sec^2(θ) - 1)dθ$, etc. It's a bit easier to use the substitution $x = \sinh(u)$ instead of the above. Tags explain, substitution, trigonometric 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 devinperry Calculus 4 April 12th, 2014 12:46 PM nephi39 Calculus 10 March 15th, 2012 08:56 AM Jet1045 Calculus 3 March 8th, 2012 11:22 AM izseekzu Calculus 1 February 16th, 2010 08:09 PM mmmboh Calculus 8 October 26th, 2008 07:46 PM

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