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Math Help - integrate!

  1. #1
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    integrate!

    Hi I have problems integrating two questions, how do I integrate sqrt(x^2-4) / x? And (4lnx) / (x(1+(lnx)^2)? Thank you very much!!
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  2. #2
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    Re: integrate!

    For the second problem, first use t= ln(x), then use z = 1+ t^2. Hope this helps!
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    Re: integrate!

    Hello, Tutu!

    \int\frac{\sqrt{x^2-4}}{x}\,dx

    It should be obvious that a Trig Substitution is called for.


    Let x \,=\,2\sec\theta \quad\Rightarrow\quad dx \,=\,2\sec\theta\tan\theta\,d\theta \quad\Rightarrow\quad \sqrt{x^2-4} \,=\,2\tan\theta

    Substitute: . \int \frac{2\tan\theta}{2\sec\theta}(2\sec\theta\tan \theta\,d\theta) \;=\;2\int\tan^2\theta\,d\theta

    . . . . . . . . =\;2\int(\sec^2-1)\,d\theta \;=\;2(\tan\theta - \theta) + C

    Now back-substitute.
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    Re: integrate!

    Quote Originally Posted by coolge View Post
    For the second problem, first use t= ln(x), then use z = 1+ t^2. Hope this helps!
    Which is, of course, the same as the single substitution z= 1+ (ln(x))^2.
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  5. #5
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    Re: integrate!

    Quote Originally Posted by HallsofIvy View Post
    Which is, of course, the same as the single substitution z= 1+ (ln(x))^2.
    My two cents. I prefer to use two separate substitutions rather than one more complicated one. I can keep better track that way. Or maybe I'm just a coward.

    -Dan
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