Evaluating definite integrals using complex contour integrals

Question

In most cases one only has to consider the complex function where we put $z$ instead $x$. For example, calculating the integral: $$\int_{0}^{\infty}\frac{dx}{x^{4}+1}$$ Here i just integrate the complex function $f\left(z\right)=\frac{dz}{z^{4}+1}$, over the first quadrant using known methods. Another example is : $$\int_{0}^{\infty}\frac{\sin^{4}x}{x^{4}}dx$$ At first I would have done the same thing as above, but a less obvious approach is to use the function(derived using Euler's identity) $$f\left(z\right)=\frac{1}{8z^{4}}\left(3-4e^{i2z}+e^{i4z}\right)$$ and integrate it over the same contour, and split the integral up into its real and imaginary part. As you see, there is no general rule of thumb here. What are some other miscellaneous functions you might have come across for which the complex analogous isn't so obvious? Maybe we can make a list here that would benefit students learning about complex analysis.