Improper Integration via Contours

Question

I've heard that one can use contour integration to calculate inconvenient improper integrals, such as in : $$\int_0^\infty \frac{1}{x^4+1}\ dx$$ In my experience with contour integration, that is, using the Residue Theorem, I'm not sure how I would get a real answer. My intuition tells me that the integration of this curve over a semicircular contour in the complex plane should yield the contour integral $\oint\frac{1}{x^4+1}\ dx = \pi i$. However, what now must I do with this integral result to calculate the improper integral at the top of this post? Thanks in advance!