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heydayana

  • 2 years ago

integrate using trig sub. problem below.

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  1. heydayana
    • 2 years ago
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    \[\int\limits_{1/2}^{1} dx/ x ^{2}\sqrt{x ^{2} + 4}\]

  2. amoodarya
    • 2 years ago
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    put x=2tan u

  3. heydayana
    • 2 years ago
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    i got as far as \[1/4 \int\limits_{1/2}^{1} \sec ^{2}\theta/ \tan ^{3}\theta+\tan ^{2}\theta \] but im not sure what to do after

  4. zepdrix
    • 2 years ago
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    After you plug everything in and factor out the constants, you should have this. \[\large \frac{1}{4}\int\limits \frac{\sec^2\theta \; d\theta}{\tan^2\theta \sqrt{\color{royalblue}{\tan^2\theta+1}}}\] I'm not sure what you did on the bottom.. hmm. From here we want to use an identity, \[\large \color{royalblue}{\tan^2\theta+1=\sec^2\theta}\]Changing the integral to,\[\large \frac{1}{4}\int\limits\limits \frac{\sec^2\theta \; d\theta}{\tan^2\theta \sqrt{\color{royalblue}{\sec^2\theta}}}\]

  5. heydayana
    • 2 years ago
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    okay im left with \[1/4\int\limits_{1/2}^{1} \sec \theta/\sec ^{2}\theta -1 \]

  6. zepdrix
    • 2 years ago
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    No you're left with,\[\large \frac{1}{4}\int\limits \frac{\sec \theta}{\tan^2\theta}d \theta\] You keep making weird substitutions.. I'm not sure why :c Oh you wanted it all in terms of secant I guess? :)

  7. zepdrix
    • 2 years ago
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    From here, it's probably better to convert everything to sines and cosines and the integral should be fairly simple from there :)

  8. heydayana
    • 2 years ago
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    yeah thats what i was trying to do. okay let me try it with sines and cosines

  9. heydayana
    • 2 years ago
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    still a bit confused as to what to do with the term thats squared

  10. zepdrix
    • 2 years ago
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    \[\large \frac{1}{4}\int\limits \frac{\sin \theta}{\cos^2\theta}d \theta\]So we get this I think? Yah it might seem a little tricky at first :) But it's just a simple `U substitution`. Let \(u=\cos \theta\).

  11. zepdrix
    • 2 years ago
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    Woops did I set that up correctly? I was trying to do the sine and cosine conversion in my head... thinking.

  12. zepdrix
    • 2 years ago
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    Yah I did that upside down :) woops lol

  13. heydayana
    • 2 years ago
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    i think its cos over sin squared

  14. zepdrix
    • 2 years ago
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    yah good call XD whatever is on the bottom is your `u` :3

  15. heydayana
    • 2 years ago
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    okay thanks a lot!

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