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Math Help - long trigo integral:

  1. #1
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    long trigo integral:

    \int {\csc^5{x}}dx

    Solutions:
    \int {\csc^3{x}}(1+\cot^2{x})dx

    \int {\csc^3{x}}dx  + \int {\cot^2{x}}{\csc^3{x}}dx

    solving for int of \int {\csc^3{x}}dx
    \frac{1}{2}(-\cot{x}\csc{x} + \ln |\csc{x} - \cot{x}|)

    this is where i got stuck for solving for \int {\cot^2{x}}{\csc^3{x}}dx :

    \int {\cot^2{x}}{\csc^3{x}}dx

    \int {\csc^5{x} - \csc^3{x} }dx
    which is repeating the csc^5{x}

    help with hints

    thanks
    Last edited by ^_^Engineer_Adam^_^; August 3rd 2007 at 05:03 AM.
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  2. #2
    Senior Member tukeywilliams's Avatar
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     \int \cot^{2}x \ \csc^{3}x \ dx = \int (\cot^{2}x )(1+\cot^{2}x) \csc x \ dx

    So  \int \cot^{2}x \csc x \ dx + \int \cot^{4}x \ \csc x \ dx
    Last edited by tukeywilliams; August 3rd 2007 at 06:48 AM.
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  3. #3
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    now confused with the integral of
    \int \cot^{4}x \ \csc x \ dx<br />

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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    now confused with the integral of
    \int \cot^{4}x \ \csc x \ dx<br />

    I don't know about anyone else, but I typically convert everything to sine and cosine functions:
    \int cot^{4}(x)csc(x)dx = \int \frac{cos^4(x)}{sin^5(x)} dx

    Do this by parts:
    u = cos(x) \implies du = sin(x) dx

    dv = \frac{cos^3(x)}{sin^5(x)} dx

    We need v.
    v = \int \frac{cos^3(x)}{sin^5(x)} dx

    v = \int \frac{cos^2(x)}{sin^5(x)} cos(x) dx

    Let z = sin(x) \implies dz = cos(x) dx
    v = \int \frac{cos^2(x)}{sin^5(x)} cos(x) dx = \int \frac{1 - z^2}{z^5} dz = \int \frac{1}{z^5} dz - \int \frac{1}{z^3}

    v = -\frac{1}{4z^4} - \frac{1}{2z^2} = - \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right )

    So back to the original integral:
    \int cot^{4}(x)csc(x)dx = \int \frac{cos^4(x)}{sin^5(x)} dx

    = - cos(x) \cdot \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right ) + \int \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right ) \cdot sin(x) dx

    Can you go from here?

    -Dan
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  5. #5
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    \int {\csc^5{x}}dx
    Here is a reduction formula, using integration by parts, which can be found in a stantard table of integrals.
    Attached Thumbnails Attached Thumbnails long trigo integral:-aug4.gif  
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  6. #6
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    thank u!
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