why we need only specific contours in complex integration? We often encounter various type of real integrals that are relatively easy to compute by converting them to complex variable functions, and then integrating them using some special contours. As for example, if we are to integrate $\displaystyle{\int_0^\infty \frac{\sin x}{x}dx}$, we take a semicircular contour with a indentation at origin, or if some integral that contains exponential terms, we use a rectangular contour, or if we integrate $\displaystyle{\int_0^\infty \frac{x^{\lambda-1}}{1+x}dx}$, where $0<\lambda<1$, we use a keyhole contour. Sometime we memorize those "special" contours and use them to integrate some "specific" functions only. My question is, why all textbooks of complex analysis use only these special contours for integrating only those functions associated with them? I mean, how do they specify or create a contour by observing only the functions they are integrating? Thanks in advance.
 A: The ones involving part of a circle or part of a line are particularly easy to parameterize, or to express in polar coordinates. Because they involve relatively simple substitutions when you actually compute the integral, they're not likely to transform a simple-looking integral into one where you say "What? How the HECK am I gonna actually compute THAT???" If you said "let me just do this integral over the topologist's sine-curve," it'd generally lead to nightmares. Even if you're not going to actually integrate, but merely estimate the integral, they're often winners, because the integrand may be particularly easy to estimate along a line of constant $r$, for instance (in polar coords) or along the $x$-axis (in rectangular coords), as one or more of the variables may drop out. 
Summary: 


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*Those contours are generally easy to express as paths

*The resulting integrals along those paths involve algebraic expressions that are not much more complicated than the original integrand

*because integrands may be simple along lines-of-constant-$u$, where $u$ is one of the coordinates in some coordinate system, they may be much easier to estimate (i.e., to say things like 'along this arc of radius $r$, the integrand is bounded by $r^2$, so the integral's absolute value is bounded by $2\pi r(r^2)$, and hence goes to $0$ as $r$ approaches zero.')

