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anonymous
 one year ago
integrate dx/(4+x^2)^2 using trig substituition. im confused i thought it had to be a square root to be used no squared can some explain how toset these up?
anonymous
 one year ago
integrate dx/(4+x^2)^2 using trig substituition. im confused i thought it had to be a square root to be used no squared can some explain how toset these up?

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zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4\[\large\rm \int\limits \frac{dx}{(4+x^2)^2}\]Ummm trig sub? Yah that'll be fun :)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4If you factor a 4 out of each term in the denominator you get\[\large\rm \int\limits\limits \frac{dx}{(4+x^2)^2}=\frac{1}{4^2}\int\limits\frac{dx}{\left(1+\frac{1}{4}x^2\right)}\]Let's bring the 1/4 into the square with the x,\[\large\rm =\frac{1}{16}\int\limits\limits\frac{dx}{\left(1+\left[\frac{x}{2}\right]^2\right)}\]Now we have that thing in the form: \(\large\rm 1+stuff^2\) So we can make the substitution: \(\large\rm stuff=\tan\theta\)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4\[\large\rm \frac{x}{2}=\tan\theta\]From there, find your \(\large\rm d\theta\). Plug in all the stuff. Then we'll use some cool trig rules to simplify everything down, and hopefully, be able to integrate easily.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4From there, find your \(\large\rm dx\)* is what I meant to say. Any trouble finding it? :o

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4Or were my steps confusing?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Im sorry I just did not understand the find dtheta part. I converted (1+tan0) to sec^20 can i take the derivative of that to dnd d0?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4\[\large\rm =\frac{1}{16}\int\limits\limits\limits\limits\frac{dx}{\left(1+\left[\color{orangered}{\frac{x}{2}}\right]^2\right)^{2}}=\frac{1}{16}\int\limits\limits\limits\limits\frac{dx}{\left(1+\color{orangered}{\tan^2\theta}\right)^{2}}=\frac{1}{16}\int\limits\frac{dx}{\sec^4\theta}\]So you've got that much figured out so far? To find dx, you need to go back to your original substitution. \(\large\rm \frac{x}{2}=\tan\theta\) Take derivative of that thing.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so it would turn do x=2tantheta and the derivative is sec^2theta do it would be dtheta over sec^2theta

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4Mmm are you missing a 2 in your dx maybe? Should look like this.\[\large\rm x=2\tan theat\qquad\to\qquad \color{royalblue}{dx=2\sec^2\theta~d\theta}\] \[\large\rm \frac{1}{16}\int\limits\limits\frac{\color{royalblue}{dx}}{\sec^4\theta}=\frac{1}{16}\int\limits\limits\frac{\color{royalblue}{2\sec^2\theta~d\theta}}{\sec^4\theta}\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4Ah I made a typo in my x :) lol

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4\[\large\rm =\frac{2}{16}\int\limits\frac{d \theta}{\sec^2\theta}\]So something like this? Ya you have the right idea.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4Recall that \[\large\rm \cos\theta=\frac{1}{\sec\theta}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah i have that! and i would just u sub right?!

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4\[\large\rm =\frac{1}{8}\int\limits \cos^2\theta~d \theta\] Hmm no. Looks like you'll have to use your `HalfAngle Formula` at this point. :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i got 1/8 * integral of cos^2theta

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[=\frac{ 1 }{ 16 }\int\limits \left( 2\cos ^2 \theta \right)d \theta=\frac{ 1 }{ 16 }\int\limits \left( 1+\cos 2\theta \right) d \theta =?\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0its going to be theta sin2theta/2 but that would make the denominator for sin 32 and it is wrong @surhithayer

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.4No, that sounds fine :) You just need your final answer in terms of \(\large\rm x\), not \(\large\rm \theta\). So you need to undo your substitution, perhaps with the use of a triangle.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits \cos 2\theta d \theta =\frac{ \sin 2\theta }{ 2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[=\frac{ 1 }{ 2 }*\frac{ 2 \tan \theta }{ 1\tan ^2 \theta }=\frac{ \tan \theta }{ 1\tan ^2 \theta }\]
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