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Here is an interesting limit of an integral I do not know how to begin. Any help is greatly appreciated.

$\lim_{n\to \infty}n\int_{0}^{\frac{\pi}{2}}(1-\sqrt[n]{\sin(x)})\,\mathrm{d}x$

I know it converges to $\frac{\pi ln(2)}{2}$, but how?.

Thanks much

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5 Answers 5

up vote 2 down vote accepted

You can integrate by parts in your integral, differentiating $1 - (\sin(x))^{1 \over n}$ and integrating 1. You get $$n\int_0^{\pi \over 2}(1 - \sin(x)^{1 \over n})\,dx = \int_0^{\pi \over 2}x \sin(x)^{{1 \over n} - 1}\cos(x)\,dx$$ (Both of the boundary terms evaluate to zero). In the right-hand integral above, the integrand increases monotonically to $x(\sin(x))^{-1}\cos(x) = x\cot(x)$ as $n$ goes to infinity. So your limit is exactly $$\int_0^{\pi \over 2}x\cot(x)\,dx$$ This is a pretty famous integral in itself, but you can integrate by parts again in the reverse direction (making sure the endpoints go to zero in the improper integral) to convert it into the integral $$-\int_0^{\pi \over 2}\ln(\sin(x))\,dx$$ As you indicated this is well known and evaluates to ${\displaystyle{\pi \ln2 \over 2}}$.

Effectively what is going on here is that you are using that the limit as $\epsilon \rightarrow 0$ of ${\displaystyle{1 - u^{\epsilon} \over \epsilon}}$ is negative the derivative of $u^x$ at $x = 0$, or $-\ln(u)$. Letting $\epsilon = {1 \over n}$ and $u = \sin(x)$ gives that your integrand (including the $n$ factor) converges to $-\ln(\sin(x))$. So your answer should be the integral of that. But making this rigorous seems a little tricky (to me) so I integrated by parts and then took limits, then integrated by parts back again.

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You should probably add a comment justifying switching limits with integration. –  Glen Wheeler Apr 22 '11 at 12:54

A trick (and it's only a trick) is to set $$f(x) = n(1-\sqrt[n]{\sin(x)})$$ so $$ \sin(x) = \left(1-\frac{f(x)}{n}\right)^n$$ which in the limit $n\to\infty$ is $$ \sin(x) = e^{-f_{\infty}(x)}$$ or $$f_{\infty}(x) = -\ln (\sin(x))$$ which you integrate to the result: $$-\int_0^{\frac{\pi}{2}}\ln(\sin(x))dx = \frac{\pi\ln 2}{2}$$

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I think you need to show $f_n$ converges uniformly, not only pointwise, to $f_\infty$ in order to switch the limit with the integral. –  Theta33 Mar 13 '11 at 18:19
1  
@Mielu: there are a lot of "holes" here that require some plugging. That's why I said it's a trick (completely valid though). –  Eelvex Mar 13 '11 at 18:32
    
Oh, I understand. –  Theta33 Mar 13 '11 at 18:42

Suggestion: substitute $u=\sin x$, and consider the Beta function.

EDIT: A better suggestion. To arrive to $$ - \int_0^{\pi /2} {\ln (\sin x)dx} \, \bigg( = \frac{{\pi \ln 2}}{2}\bigg), $$ consider $$ \mathop {\lim }\limits_{r \to \infty } \frac{{\int_0^{\pi /2} {[1 - (\sin x)^{1/r} ]dx} }}{{1/r}} = \mathop {\lim }\limits_{r \to \infty } \frac{{\int_0^{\pi /2} {[1 - e^{\ln (\sin x)/r} ]dx} }}{{1/r}}, $$ followed by L'Hospital's rule.

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I meant that for $\int_0^{\pi /2} {(\sin x)^{1/n} dx} $. –  Shai Covo Mar 13 '11 at 11:00
    
Thank you. I ended up taking the limit and getting $-\int_{0}^{\frac{\pi}{2}}ln(sin(x))dx$. A rather famous integral I can do. This evaluates to the correct solution. –  Cody Mar 13 '11 at 11:03

Since $$ \lim_{n\to\infty}n\left(x^{1/n}-1\right)=\log(x)\tag{1} $$ converges monotonically, $$ \begin{align} \lim_{n\to\infty}n\int_0^{\pi/2}\left(1-\sqrt[\large n]{\sin(x)}\right)\,\mathrm{d}x &=-\int_0^{\pi/2}\log(\sin(x))\,\mathrm{d}x\\ &=-\frac12\int_0^\pi\log(\sin(x))\,\mathrm{d}x\\ &=-\int_0^{\pi/2}\log(\sin(2x))\,\mathrm{d}x\\ &=-\int_0^{\pi/2}\Big(\log(2)+\log(\sin(x))+\log(\cos(x))\Big)\,\mathrm{d}x\\ &=-\frac\pi2\log(2)-2\int_0^{\pi/2}\log(\sin(x))\,\mathrm{d}x\tag{2} \end{align} $$ Solving $(2)$, we get $$ -\int_0^{\pi/2}\log(\sin(x))\,\mathrm{d}x=\frac\pi2\log(2)\tag{3} $$

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$$n\int_0^{\frac{\pi}{2}}(1-\sqrt[n]{\sin(x)})\mathrm dx=\pi\left(\frac{n}{2}-n^2 2^{1-\frac1{n}}\frac{\Gamma\left(\frac1{n}\right)}{\Gamma\left(\frac1{2n}\right)^2}\right)$$

$$\approx\pi\left(\ln\sqrt{2}-\left(\left(\ln\sqrt{2}\right)^2+\frac{\pi^2}{48}\right)\frac1{n}+O\left(\frac1{n^2}\right)\right)$$

Now, take the limit...

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how do you get the approximation $\approx\pi(\ln{\sqrt{2}}...$ ? –  Weltschmerz Mar 13 '11 at 13:17
    
I don't understand either of the two steps. –  Dan Petersen Mar 13 '11 at 15:29
    
Thank everyone very, very much for your input. I always learn from you all :) –  Cody Mar 15 '11 at 9:56

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