Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$ I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$
Which function/ contour should I consider ?

First,you may use the definition of Lapace transform to get it or integration by parts twice.
For complex numbers your integrand is the real part of $\exp(-ax+ibx)$. Use this function to evaluate the unbounded integral then evaluate the bounded one 
