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Math Help - Trigometric Integrals

  1. #1
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    Trigometric Integrals

    Evaluate the indefinite integral.


    so first i simplified it to ∫(1/4√x-16)x dx
    i took the integral by letting x = 4 secθ, dx= 4secθtanθ
    then for √x-16)= 4tanθ
    then i pluged in √x-16)= 4tanθ and x= 4 secθ for the equation ∫(1/4√x-16)x dx and my intergral came out to be -1/4cosθ + C but it is wrong i dont get what i have to do next?

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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by emurphy View Post
    Evaluate the indefinite integral.


    so first i simplified it to ∫(1/4√x-16)x dx
    i took the integral by letting x = 4 secθ, dx= 4secθtanθ
    then for √x-16)= 4tanθ
    then i pluged in √x-16)= 4tanθ and x= 4 secθ for the equation ∫(1/4√x-16)x dx and my intergral came out to be -1/4cosθ + C but it is wrong i dont get what i have to do next?

    \int\frac{\sqrt{16x^2-256}}{x}\,dx=4\int\frac{\sqrt{x^2-16}}{x}\,dx

    You made a correct choice for x: x=4\sec t, dx=4\sec t\tan t\,dt

    4\int\frac{4\tan t}{4\sec t}\cdot4\sec t\tan t\,dt=16\int\tan^2t\,dt=16\int(\sec^2t-1)\,dt=16\tan t-16t

    Now sub back in for t.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by emurphy View Post
    Evaluate the indefinite integral.


    so first i simplified it to ∫(1/4√x-16)x dx
    i took the integral by letting x = 4 secθ, dx= 4secθtanθ
    then for √x-16)= 4tanθ
    then i pluged in √x-16)= 4tanθ and x= 4 secθ for the equation ∫(1/4√x-16)x dx and my intergral came out to be -1/4cosθ + C but it is wrong i dont get what i have to do next?


    {4}\int \frac{\sqrt{x^2-16}}{x} \ dx = 4\int \sqrt{1-16/x^2} \ dx

    Set 4/x=\sin \theta, dx = -4\cot \theta \csc \theta\ d\theta

    4\int \sqrt{1-\sin^2 \theta} (-4\cot \theta \csc \theta\ d\theta) = -16\int \cos \theta \cot \theta \csc \theta\ d\theta

    =-16 \int \cos \theta \frac{\cos \theta}{\sin \theta}\frac{1}{\sin \theta}\ d\theta = -16 \int  \cot^2 \theta d\theta

    =16 (\theta+\cot \theta)+ C = 16(\arcsin(4/x)+\cot(\arcsin(4/x)))+C

    Edit : above is better!
    Last edited by Bruno J.; October 19th 2009 at 07:15 PM.
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  4. #4
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    I dont understand how to get t
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  5. #5
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by emurphy View Post
    I dont understand how to get t
    If x=4\sec t, then t=\sec^{-1}\left(\frac{x}{4}\right)
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