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Another integral

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Prove that for \(\displaystyle n >1\) we have

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

This appeared in the Bee integration contest.
 

Random Variable

Well-known member
MHB Math Helper
Jan 31, 2012
253
$ \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} $
 

chisigma

Well-known member
Feb 13, 2012
1,704
Prove that for \(\displaystyle n >1\) we have

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

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

$\chi$ $\sigma$
 

Pranav

Well-known member
Nov 4, 2013
428
Prove that for \(\displaystyle n >1\) we have

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

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:

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