MHB Integral Prove: $\frac {n}{n^{2} - 1}$ for $n > 1$

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The integral $$\int^{\infty}_{0}\frac {dx}{\left(x + \sqrt {1 + x^{2}}\right)^{n}}$$ for $$n > 1$$ is equal to $$\frac {n}{n^{2} - 1}$$, as posed in the Bee integration contest. Participants are tasked with proving this equality through various mathematical techniques. The discussion emphasizes the importance of rigorous proof and explores different approaches to tackle the integral. The problem has generated interest due to its complexity and relevance in advanced calculus. Engaging with this integral can enhance understanding of integration techniques and their applications.
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Prove that for $$n >1$$ we have

$$\int^{\infty}_{0}\frac {dx}{\left(x + \sqrt {1 + x^{2}}\right)^{n}} = \frac {n}{n^{2} - 1}$$​

This appeared in the Bee integration contest.
 
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$ \displaystyle \int_{0}^{\infty} \frac{1}{(x+\sqrt{1+x^{2}})^{n}} \ dx $Let $u = \sinh x $.$ \displaystyle = \int_{0}^{1} \frac{1}{(\sinh x + \cosh x)^{n}} \ \cosh x \ dx = \int_{0}^{\infty} e^{-nx} \cosh x \ dx $

$ \displaystyle = \frac{1}{2} \int_{0}^{\infty} e^{-(n-1)x} \ dx + \frac{1}{2} \int_{0}^{\infty} e^{-(n+1)x} \ dx $

$ \displaystyle = \frac{1}{2} \frac{1}{n-1} + \frac{1}{2} \frac{1}{n+1} = \frac{n}{n^{2}-1} $
 
ZaidAlyafey said:
Prove that for $$n >1$$ we have

$$\int^{\infty}_{0}\frac {dx}{\left(x + \sqrt {1 + x^{2}}\right)^{n}} = \frac {n}{n^{2} - 1}$$​

This appeared in the Bee integration contest.
Setting $\displaystyle x = \sinh t$ the integral becomes...

$\displaystyle I = \int_{0}^{\infty} \frac{\cosh t}{(\sinh t + \cosh t)^{n}}\ dt\ (1)$ ... and because is...

$\displaystyle \int \frac{\cosh t}{(\sinh t + \cosh t)^{n}}\ dt = - \frac{(\sinh t + \cosh t)^{- n}\ (n \ \cosh t + \sinh t)}{n^{2}-1}\ (2)$

... the result is immediate...

Kind regards

$\chi$ $\sigma$
 
ZaidAlyafey said:
Prove that for $$n >1$$ we have

$$\int^{\infty}_{0}\frac {dx}{\left(x + \sqrt {1 + x^{2}}\right)^{n}} = \frac {n}{n^{2} - 1}$$​

This appeared in the Bee integration contest.

Hello again! :)

Use the substitution $x+\sqrt{1+x^2}=e^t$. A bit of simplification gives $\displaystyle x=\frac{e^t-e^{-t}}{2} \Rightarrow dx=\frac{e^t+e^{-t}}{2}dt$

Hence, the integral is

$$\int_0^{\infty} \frac{e^t+e^{-t}}{2e^{nt}}dt$$

$$=\frac{1}{2}\int_0^{\infty}\left( e^{-t(n-1)}+e^{-(n+1)}\right)dt$$
Its easy to solve the above integral so I skip the steps. Solving the integral gives:

$$\frac{1}{2}\left(\frac{1}{n+1}+\frac{1}{n-1}\right)=\frac{n}{n^2-1}$$
 
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