How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help.

$$
\begin{aligned}
\int_0^{\infty} \frac{d t}{1+t^2}&=\int_0^{\infty} \frac{\frac{1}{t^2}}{t^2+\frac{1}{t^2}} d t \\=& \frac{1}{2} \int_0^{\infty} \frac{\left(1+\frac{1}{t^2}\right)-\left(1-\frac{1}{t^2}\right)}{t^2+\frac{1}{t^2}} d t \\
=& \frac{1}{2}\left[\int_0^{\infty} \frac{d\left(t-\frac{1}{t}\right)}{\left(t-\frac{1}{t}\right)^2+2}-\int_0^{\infty} \frac{d\left(t+\frac{1}{t}\right)}{\left(t+\frac{1}{t}\right)^2-2}\right]_0^{\infty} \\
=& \frac{1}{2 \sqrt{2}}\left[\tan ^{-1}\left(\frac{t-\frac{1}{t}}{\sqrt{2}}\right)\right]_0^{\infty}-\frac{1}{2 \sqrt{2}}\left[\ln \left|\frac{t+\frac{1}{t}-\sqrt{2}}{t+\frac{1}{t}+\sqrt{2}} \right|\right]_0^{\infty}\\
=& \frac{\pi}{2 \sqrt{2}}
\end{aligned}
$$
