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Example 1

Consider the integral ∫ − ∞ ∞ 1 (x 2 + 1) 2 d x, {\displaystyle \int _{-\infty }^{\infty }{\frac {1}{\left(x^{2}+1\right)^{2}}}\,dx,}

To evaluate this integral, we look at the complex-valued function f (z) = 1 (z 2 + 1) 2 {\displaystyle f(z)={\frac {1}{\left(z^{2}+1\right)^{2}}}}

which has singularities at i and − i. We choose a contour that will enclose the real-valued integral, here a semicircle with boundary diameter on the real line (going from, say, − a to a) will be convenient. Call this contour C.

There are two ways of proceeding, using the Cauchy integral formula or by the method of residues: