How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?

Solution 1:

Note that the substitution $t=1/u$ changes the integral to $$\int_0^\infty \frac{u^2}{1+u^4}du.$$ Doesn't sound very helpful, but there was extensive discussion of that here.

Added: But we can do better. Split the integral into into two parts, $0$ to $1$, and $1$ to infinity. On the second part, let $t=1/u$. We get $\int_0^1 \frac{u^2}{1+u^4}du$. Now $u$ has done its duty, and is discarded for the more popular $t$. Our original integral is equal to $$\int_0^1 \frac{1+t^2}{1+t^4} dt.$$

There is now a minor miracle. $$\frac{1}{1-\sqrt{2}t+t^2}+ \frac{1}{1+\sqrt{2}t+t^2}=\frac{2(1+t^2)}{1+t^4}.$$

Complete the square(s) as usual.

Solution 2:

If you don't want to do a partial fraction decomposition, put $I:=\int_0^{+\infty}\frac{dt}{1+t^4}$. By the substitution $x=\frac 1t$ on $\left[0,+\infty\right)$ and $\left(-\infty,0\right]$, we get $2I = \int_{-\infty}^{+\infty}\frac{x^2}{1+x^4}\,dx$. Now we have \begin{align*} 4I &= \int_{-\infty}^{+\infty}\frac{t^2-\sqrt 2t+1}{(t^2-\sqrt 2t+1)(t^2+\sqrt 2t+1)}\,dt \\\ &=\int_{-\infty}^{+\infty}\frac{1}{t^2+\sqrt 2t+1}\,dt\\\ &=\int_{-\infty}^{+\infty}\frac{1}{(t+\frac{\sqrt 2}2)^2-\frac 12+1}\,dt\\\ &=\int_{-\infty}^{+\infty}\frac{du}{u^2+\frac 12}\\\ &=2\int_{-\infty}^{+\infty}\frac{du}{(\sqrt 2 u)^2+1}\\\ &=\frac 2{\sqrt 2}\arctan (\sqrt 2u)\mid_{u=-\infty}^{u=+\infty}\\\ &=\frac {2\pi}{\sqrt 2} \end{align*} and finally $I=\dfrac{\pi}{2\sqrt 2}$.

Solution 3:

You can also do it by partial fraction decomposition. We have $$ \frac{2\sqrt{2}}{1+t^4} = \frac{t + \sqrt{2}}{t^2 + \sqrt{2}t + 1} - \frac{t - \sqrt{2}}{t^2 - \sqrt{2}t + 1}. $$ We have $$\int_0^\infty \frac{dt}{t^2 \pm \sqrt{2}t + 1} = \int_0^\infty \frac{dt}{(t \pm \sqrt{2}/2)^2 + 1/2} = \int_0^\infty \frac{2dt}{(\sqrt{2} t \pm 1)^2 + 1} = \sqrt{2} \arctan (\sqrt{2}t \pm 1) \big|_0^\infty.$$ Continuing, we get $$\frac{\sqrt{2}\pi}{2} - \sqrt{2} \arctan (\pm 1) = \frac{\sqrt{2}\pi}{2} \mp \frac{\sqrt{2}}{4} = \frac{\sqrt{2}\pi (2 \mp 1)}{4}.$$ Next, we have $$ \int_0^\infty \frac{(2t \pm \sqrt{2}) dt}{t^2 \pm \sqrt{2}t + 1} = \log (t^2 \pm \sqrt{2}t + 1) \big|_0^\infty. $$ The integral doesn't converge, but we can consider instead $$ \int_0^\infty \frac{(2t + \sqrt{2}) dt}{t^2 + \sqrt{2}t + 1} - \frac{(2t - \sqrt{2}) dt}{t^2 - \sqrt{2}t + 1} = \log \frac{t^2 + \sqrt{2}t + 1}{t^2 - \sqrt{2}t + 1} \big|_0^\infty = 0. $$ Therefore we rewrite our initial expansion $$ \frac{4\sqrt{2}}{1+t^4} = \frac{2t + 2\sqrt{2}}{t^2 + \sqrt{2}t + 1} - \frac{2t - 2\sqrt{2}}{t^2 - \sqrt{2}t + 1} = \frac{2t + \sqrt{2}}{t^2 + \sqrt{2}t + 1} - \frac{2t - \sqrt{2}}{t^2 - \sqrt{2}t + 1} + \frac{\sqrt{2}}{t^2 + \sqrt{2}t + 1} + \frac{\sqrt{2}}{t^2 - \sqrt{2}t + 1}. $$ Integrating, we get $$ \int_0^\infty \frac{4\sqrt{2}}{1+t^4} = \sqrt{2} \frac{\sqrt{2} \pi (2-1)}{4} + \sqrt{2} \frac{\sqrt{2} \pi (2+1)}{4} = \frac{\pi}{2} + \frac{3\pi}{2} = 2\pi. $$ Therefore the integral we want is $$ \int_0^\infty \frac{1}{1 + t^4} = \frac{2\pi}{4\sqrt{2}} = \frac{\sqrt{2}\pi}{4}. $$

Solution 4:

$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ With $\ds{x \equiv {1 \over 1 + t^{4}} \quad\iff\quad t = \pars{1 - x \over x}^{1/4}}$: \begin{align} \color{#00f}{\large\int_{0}^{\infty}{\dd t \over 1 + t^{4}}}&= \int_{1}^{0}x\,{1 \over 4}\,\pars{1 - x \over x}^{-3/4} \pars{-\,{1 \over x^{2}}}\,\dd x = {1 \over 4}\int_{0}^{1}x^{-1/4}\pars{1 - x}^{-3/4}\,\dd x \\[3mm]&= {1 \over 4}\,{\rm B}\pars{{3 \over 4},{1 \over 4}} ={1 \over 4}\,{\Gamma\pars{3/4}\Gamma\pars{1/4} \over \Gamma\pars{3/4 + 1/4}} ={1 \over 4}\,{\pi \over \sin\pars{\pi\bracks{1/4}}} = \color{#00f}{\large{\root{2} \over 4}\,\pi} \approx 1.1107 \end{align}