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Math Help - integral limit

  1. #1
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    integral limit

    Calculate :
    \lim_{n\rightarrow \infty} \displaystyle\int^ \frac{\pi}{2}_0 \frac{\sqrt{3} }{ctg^nx+tg^nx} \,dx
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by petter View Post
    Calculate :
    \lim_{n\rightarrow \infty} \displaystyle\int^ \frac{\pi}{2}_0 \frac{\sqrt{3} }{ctg^nx+tg^nx} \,dx
    Consider that \left|\frac{1}{\cot^n(x)+\tan^n(x)}\right|\leqslan  t{1}~\forall{x}\in\left(0,\frac{\pi}{2}\right), and since \int_0^{\frac{\pi}{2}}dx converges, we can state that \int_0^{\frac{\pi}{2}}\frac{\sqrt{3}}{\cot^n(x)+\t  an^n(x)}dx converges uniformly. So \lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\frac{\sqrt  {3}}{\cot^n(x)+\tan^n(x)}=\int_0^{\frac{\pi}{2}}\l  im_{n\to\infty}\frac{\sqrt{3}}{\cot^n(x)+\tan^n(x)  }dx. Can you go from there?

    EDIT: Ill finish it. \forall{x}\in\left(0,\frac{\pi}{4}\right)~\cot(x)>  1

    And \forall{x}\in\left(\frac{\pi}{4},\frac{\pi}{2}\rig  ht)~\tan(x)>1

    So rewrite this integral as \int_0^{\frac{\pi}{4}}\lim_{n\to\infty}\frac{\sqrt  {3}}{\cot^n(x)+\tan^n(x)}dx+\int_{\frac{\pi}{4}}^{  \frac{\pi}{2}}\lim_{n\to\infty}\frac{\sqrt{3}}{\co  t^n(x)+\tan^n(x)}dx

    Now since on those intervals one of the functions is greater than one we have that both limits go to zero.

    Thus \lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\frac{\sqrt  {3}}{\cot^n(x)+\tan^n(x)}dx=0
    Last edited by Mathstud28; November 27th 2008 at 08:47 AM.
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