The evaluation of the definite integrals $$\displaystyle \int_{0}^{\pi}\lfloor \cot x \rfloor dx$$ and $$\displaystyle \int_{0}^{\pi}\lfloor \cos x \rfloor dx$$ utilizes the floor function, denoted as $$\lfloor x \rfloor$$. The integral $$\displaystyle \int_{0}^{\pi} \lfloor \cot x \rfloor dx$$ is transformed into a telescopic sum, resulting in a final value of $$-\frac{\pi}{2}$$. This approach also includes the equivalence of the cotangent and tangent functions across specified intervals, demonstrating the improper nature of the integral.
PREREQUISITESMathematicians, calculus students, and educators interested in advanced integral evaluation techniques, particularly those involving the floor function and trigonometric identities.
jacks said:Evaluation of $$\displaystyle \int_{0}^{\pi}\lfloor \cot x \rfloor dx$$ and $$\displaystyle \int_{0}^{\pi}\lfloor \cos x \rfloor dx\;,$$ where $$\lfloor x \rfloor $$ denote Floor function of $$x$$