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Math Help - integral

  1. #1
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    integral

    how do i integrate:
    \int{cot^2(x)} dx
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  2. #2
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    Quote Originally Posted by scorpion007 View Post
    how do i integrate:
    \int{cot^2(x)} dx
    Hi,

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

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

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

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

    \int{\left( \frac{1}{sin^2(x)}-1\right)} dx =-cot(x)-x+C

    EB
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  3. #3
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    Quote Originally Posted by earboth View Post
    Hi,

    \int{cot^2(x)} dx = \int{\frac{cos^2(x)}{sin^2(x)}} dx
    From here you are going around the houses to get where you are going:

    \int{\frac{\cos^2(x)}{\sin^2(x)}} dx= \int{\frac{1-\sin^2(x)}{\sin^2(x)}} dx=\int{\frac{1}{\sin^2(x)}-1} dx

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

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

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

    \int{\left( \frac{1}{sin^2(x)}-1\right)} dx =-cot(x)-x+C

    EB
    But you are using the standard integral:

    \int \csc^2(x) dx =-\cot(x) ,

    in which case you just could have used the sandard integral:

    \int{\cot^2(x)} dx =-\cot(x)-x.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    From here you are going around the houses to get where you are going:...

    RonL
    It's so much fun to go the long way round...

    Actually

    - the functions sec, csc, cot were not taught in German schools because you can express them by the standard trig functions;

    - I was very proud about me that I detected the solution I wrote to scorpion;

    - you are right.

    EB
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  5. #5
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    Hello, scorpion007!

    \int \cot^2(x)\,dx

    Use the identity: . \cot^2\theta\:=\:\csc^2\theta - 1

    \int\cot^2x\,dx\;=\;\int\left(\csc^2x - 1\right)\,dx\;=\;-\cot x - x + C


    Edit: corrected the typo.
    . . . .Thanks for the heads-up, Scorpion!
    Last edited by Soroban; October 26th 2006 at 08:18 PM.
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  6. #6
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    thanks very much guys.

    (Soroban, you made a little typo, you wrote cos instead of cot in your last expression.)
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