Dirichlet's Lemma

$$\int_0^\pi {\sin[(n+{\textstyle{1\over 2}})x]\over 2\sin({\textstyle{1\over 2}}x)}\,dx={\textstyle{1\over 2}}\pi,$$

where the Kernel is the Dirichlet Kernel.

References

Cohn, H. Advanced Number Theory. New York: Dover, p. 37, 1980.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1101, 1979.