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# An integral with trigonometric and rational function

Evaluate the integral

Solution

Firstly , we begin by noticing the following

Let us now consider the function as well as the contour

The function has three simple poles of which only is included within the contour. The residue at turns out to be . Thus

The contribution of the large circle as is whereas the contribution of the small circle as is . This can be seen by parametrising the small circle ( ). Hence:

Using we get that