Integration of $\int_{0}^{\infty}\frac{dw}{1+\left ( \frac{w}{B} \right )^4}$ with the help of signal properties. 
Find the integral
  $$\int_{0}^{\infty}\frac{dw}{1+\left ( \frac{w}{B} \right )^4}$$
  where $B$ is a constant.

This integration i tried by normal method that gives the result $\frac{\pi B}{2\sqrt2}$ but it goes so lengthy like this

is there any signal properties that i use to solve it in simple steps?


With substitution $x^4=t$ and again with $\dfrac{1}{1+t}=u$ we have
\begin{align}
I 
&= B\int_0^\infty\dfrac{1}{1+x^4}dx \\
&= \dfrac{B}{4}\int_0^\infty\dfrac{t^{-\frac34}}{1+t}dt \\
&= \dfrac{B}{4}\int_0^1 u^{-\frac14}(1-u)^{-\frac34}du\\
&= \dfrac{B}{4}\beta(\frac14,\frac34) \\
&= \dfrac{B}{4}\dfrac{\Gamma(\dfrac14)\Gamma(\dfrac34)}{\Gamma(1)} \\
&= \dfrac{B}{4}\dfrac{\pi}{\sin\frac14\pi} \\
&= \color{blue}{\dfrac{\pi B}{2\sqrt{2}}}
\end{align}
where $\beta(x,y)$ is beta function.
