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Math Help - integral

  1. #1
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    integral

    I have a problem i dont have the solution manual too and i am stumped on how to complete.

    This could be a trigonometric substitution?

    \int_1^2 \frac {dx}{x \sqrt {4 + x^2}}

    this is what i came up with but i dont think its right..

     x = 2 tan \theta

     dx = 2 sec^2 \theta

    and

     \sqrt {2^2 + ( 2tan \theta)2}}

    identity
    (tan^2\theta -1)2^2

    = 2sec\theta

    substitute back in to the original equation..

    \int_1^2 \frac{2 sec^2 \theta} {2 tan\theta 2 sec\theta}2 sec^2\theta

    cross multiply


    \int_1^2 \frac{2 sec^2 \theta} {2 tan\theta}

    not sure if the 2 can cross out or what to do with them or if i am even right at this point?
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    Re: integral

    \int_1^2 \frac{dx}{x\sqrt{4+x^2}}

    x = 2\tan{t}

    dx = 2\sec^2{t} \, dt

    \int_{\arctan(1/2)}^{\pi /4} \frac{2\sec^2{t}}{2\tan{t} \cdot 2\sec{t}} \, dt

    \frac{1}{2} \int_{\arctan(1/2)}^{\pi /4} \frac{\sec{t}}{\tan{t}} \, dt

    \frac{1}{2} \int_{\arctan(1/2)}^{\pi /4} \csc{t} \, dt

    finish it?
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  3. #3
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    Re: integral

    where did you get the arctan and pi from?
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  4. #4
    Super Member TheChaz's Avatar
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    Re: integral

    When you do a substitution, you must change the limits of integration.
    Some find it easier to just leave the limits blank until back-substituting towards the end.
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