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Math Help - cot(x) and ln sin(x)

  1. #1
    Member vernal's Avatar
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    cot(x) and ln sin(x)

    Hi, please help me, what can prove it

    cot(x)  and ln sin(x)-untitled.jpg

    thanks
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  2. #2
    Member Sylvia104's Avatar
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    Re: cot(x) and ln sin(x)

    Try \int_{\frac{\pi}2}^xf(t)\,\mathrm dt \leqslant \int_{\frac{\pi}2}^x\cot t\,\mathrm dt \leqslant \int_{\frac{\pi}2}^xg(t)\,\mathrm dt for some functions f(t),g(t).

    By the way, you stated the range of x as x\leq\frac{\pi}2\leq\pi but I think you mean \frac\pi2\leq x<\pi.

    PS: f(t)=\frac{\mathrm d}{\mathrm dt}\left[t\cot t\right] and g(t)\equiv0.
    Last edited by Sylvia104; May 17th 2012 at 08:51 AM.
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  3. #3
    Member Sylvia104's Avatar
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    Re: cot(x) and ln sin(x)

    I beg your pardon, the functions are negative in the given range so we need to do it this way:

    Show that 0\leqslant -\cot t \leqslant -f(t) for \frac{\pi}2\leqslant t<\pi where f(t) = \frac{\mathrm d}{\mathrm dt}\left[t\cot t\right] = \cot t-t\csc^2t.

    Then integrate with respect to t from t=\frac{\pi}2 to t=x and multiply through by -1.
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  4. #4
    Member vernal's Avatar
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    Re: cot(x) and ln sin(x)

    Quote Originally Posted by Sylvia104 View Post
    Try \int_{\frac{\pi}2}^xf(t)\,\mathrm dt \leqslant \int_{\frac{\pi}2}^x\cot t\,\mathrm dt \leqslant \int_{\frac{\pi}2}^xg(t)\,\mathrm dt for some functions f(t),g(t).

    By the way, you stated the range of x as x\leq\frac{\pi}2\leq\pi but I think you mean \frac\pi2\leq x<\pi.

    PS: f(t)=\frac{\mathrm d}{\mathrm dt}\left[t\cot t\right] and g(t)\equiv0.

    Sorry. yes .

    thanks
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  5. #5
    Member vernal's Avatar
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    Re: cot(x) and ln sin(x)

    thanks, but i can't understand. There another way?
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