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Math Help - Need help for an integral

  1. #1
    MHF Contributor arbolis's Avatar
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    Need help for an integral

    I didn't recall
    but I don't think my result is that strange or difficult to compute.

    The integral is \int \frac{dx}{x\sqrt{1-x^2}}. I'm sorry if I posted it before, but I don't think so (even in that case, I would now solve it differently).
    I made the substitution x=\sin(\theta) and I could reach that the integral's worth \int \cot (\theta )\cdot \sec(\theta)d\theta. I don't know how to solve this one...
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  2. #2
    Super Member wingless's Avatar
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    \int \frac{dx}{x\sqrt{1-x^2}} = \int \frac{x~dx}{x^2\sqrt{1-x^2}}

    Let u = \sqrt{1-x^2}, then u^2 = 1-x^2 and x~dx = -u~du.

    \int \frac{-\not u~du}{(1-u^2)\not u} = -\int \frac{1}{1-u^2} ~du. I think you can finish it from here.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    \int\frac{dx}{x\sqrt{1-x^2}}

    Let x=\sin(\theta)

    So

    dx=\cos(\theta)~d\theta

    So we have

    \int\frac{\cos(\theta)}{\sin(\theta)\sqrt{1-\sin^2(\theta)}}~d\theta

    =\int\csc(\theta)

    =-\ln|\csc(\theta)+\cot(\theta)|=\ln\left|\tan\left(  \frac{x}{2}\right)\right|

    \underbrace{=}_{\text{backsub}}\ln\left|\tan\left(  \frac{\arcsin(x)}{2}\right)\right|
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  4. #4
    MHF Contributor arbolis's Avatar
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    Mathstud28, you wrote
    =-\ln|\csc(\theta)+\cot(\theta)|=\ln\left|\tan\left(  \frac{x}{2}\right)\right|
    , shouldn't it be =-\ln|\csc(\theta)+\cot(\theta)|=\ln\left|\tan\left(  \frac{\theta}{2}\right)\right|? Also, I understand everything you did but not this step. Could you detail a bit more this step? I also guess that =\int\csc(\theta) is a value I MUST know, right?
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by arbolis View Post
    Mathstud28, you wrote , shouldn't it be =-\ln|\csc(\theta)+\cot(\theta)|=\ln\left|\tan\left(  \frac{\theta}{2}\right)\right|? Also, I understand everything you did but not this step. Could you detail a bit more this step? I also guess that =\int\csc(\theta) is a value I MUST know, right?
    Yeah it should be theta not x

    And this is one you should know.
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  6. #6
    MHF Contributor arbolis's Avatar
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    Thanks Mathstud28! I could reach =-\ln|\csc(\theta)+\cot(\theta)|=\ln\left|\tan\left(  \frac{\theta}{2}\right)\right|
    And wingless,


    Let , then and .

    . I think you can finish it from here.
    Very nice way to solve it! The problem wanted a trig or hyperbolic sub, but I should train myself to any way, so I'll try it either. Any prob I get, I post here.
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