Using Cauchy theorem to compute complicated real integrals

As the central object in theory of complex analysis, Holomorphic functions have many elegant mathematical properties. are extremely valuable because, on one hand, they unexpectedly common, and other may be used to establish powerful theorems. For example, To theorems like prime number theorem, analytic theorists commonly create holomorphic or meromorphic that hold number-theoretic information, such as Riemann zeta function. Given knowledge about a function relatively small part its domain, extract information function's behavior priori unrelated sections according Cauchy's integral formula identity theorem (and this is what allows things contour integration work). Due difficulties obtaining primitives some real function, fundamental Calculus does not work most cases. In article, we review Cauchy use it tool compute several integrals.