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Math Help - Integration of 1/sin^4(x)

  1. #1
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    Integration of 1/sin^4(x)

    I need help integrating this function. I've done integration by parts many times, and it seems to be an infinite loop. Please help, would be greatly appreciated!
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    Re: Integration of 1/sin^4(x)

    Quote Originally Posted by mathimaticool View Post
    I need help integrating this function. I've done integration by parts many times, and it seems to be an infinite loop. Please help, would be greatly appreciated!
    \frac{1}{\sin^4(x)}=\csc^4(x)=\csc^2(x)\csc^2(x)=(  1+\cot^2(x))\csc^2(x)

    Now let u=\cot(x) \implies du=-\csc^2(x)dx

    Can you finish from here?
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    Re: Integration of 1/sin^4(x)

    Ahh ok. But when I get to

    -cot^2(x) - integral(csc^2(x)cot(x) dx)

    I don't know what to do. Thanks a lot
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    Re: Integration of 1/sin^4(x)

    Quote Originally Posted by mathimaticool View Post
    Ahh ok. But when I get to

    -cot^2(x) - integral(csc^2(x)cot(x) dx)

    I don't know what to do. Thanks a lot
    So you have

    \int (1+\cot^2(x))\csc^2(x) dx

    If you let u=\cot(x) \implies du = -\csc^2(x)

    This gives

    \int (1+\cot^2(x))\csc^2(x) dx=\int (1+u^2)(-du)=-\int (1+u^2)du
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    Re: Integration of 1/sin^4(x)

    AHHH thanks soo much! Got it!
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