# A integral problem hard to solve

• Mar 30th 2009, 06:46 AM
mymaydayya
A integral problem hard to solve
I am stuck for weeks...

integral from 0 to infinity 1/(1+x^(2n)) dx

someone told me tips... but I still can't solve

tips:x^n=tan

Could anyone teach me how to solve it step by step?
• Mar 30th 2009, 07:06 AM
NonCommAlg
Quote:

Originally Posted by mymaydayya
I am stuck for weeks...

integral from 0 to infinity 1/(1+x^(2n)) dx

someone told me tips... but I still can't solve

tips:x^n=tan

Could anyone teach me how to solve it step by step?

let $\displaystyle x=(\tan t)^{\frac{1}{n}}.$ then $\displaystyle dx=\frac{1}{n}(1 + \tan^2 t)(\tan t)^{\frac{1}{n} - 1}dt.$ therefore:

$\displaystyle \int_0^{\infty} \frac{dx}{1+x^{2n}}=\frac{1}{n}\int_0^{\frac{\pi}{ 2}}(\tan t)^{\frac{1}{n} - 1} dt=\frac{1}{n} \int_0^{\frac{\pi}{2}} (\sin t)^{\frac{1}{n}-1}(\cos t)^{1 - \frac{1}{n}} dt$

$\displaystyle =\frac{1}{2n}B \left(\frac{1}{2n}, 1- \frac{1}{2n} \right)=\frac{1}{2n} \Gamma \left(\frac{1}{2n} \right) \Gamma \left(1 - \frac{1}{2n} \right) = \frac{\pi}{2n \sin \left(\frac{\pi}{2n} \right)}.$

the last identity is true by Euler's reflection formula.
• Mar 30th 2009, 07:21 AM
mymaydayya
Quote:

Originally Posted by NonCommAlg
let $\displaystyle x=(\tan t)^{\frac{1}{n}}.$ then $\displaystyle dx=\frac{1}{n}(1 + \tan^2 t)(\tan t)^{\frac{1}{n} - 1}dt.$ therefore:

$\displaystyle \int_0^{\infty} \frac{dx}{1+x^{2n}}=\frac{1}{n}\int_0^{\frac{\pi}{ 2}}(\tan t)^{\frac{1}{n} - 1} dt=\frac{1}{n} \int_0^{\frac{\pi}{2}} (\sin t)^{\frac{1}{n}-1}(\cos t)^{1 - \frac{1}{n}} dt$

$\displaystyle =\frac{1}{2n}B \left(\frac{1}{2n}, 1- \frac{1}{2n} \right)=\frac{1}{2n} \Gamma \left(\frac{1}{2n} \right) \Gamma \left(1 - \frac{1}{2n} \right) = \frac{\pi}{2n \sin \left(\frac{\pi}{2n} \right)}.$

the last identity is true by Euler's reflection formula.

sorry I don't know
"B(1/2n,1-1/2n)" and the next one
stand for
could you give me link to wiki about it?
• Mar 30th 2009, 10:32 AM
NonCommAlg
Quote:

Originally Posted by mymaydayya
sorry I don't know
"B(1/2n,1-1/2n)" and the next one
stand for
could you give me link to wiki about it?

they are called "gamma" and "beta" functions. google them!