About contour integration and line integral I wanted to know why the integral of a function $f:\Omega\subset\mathbb{C}\mapsto\mathbb{C}$ over a contour $\gamma:[a,b]\mapsto \Omega $ given by $\int_a^bf(\gamma(t))\gamma’(t)dt$ and not $\int_a^bf(\gamma(t))|\gamma’(t)|dt$. Because if yes then the latter definition is more intutive as because if $f:\Omega\subset\mathbb{C}\mapsto\mathbb{R}$ then the definition of the contour integral will coincide with the definition of line integral of a scalar valued function over a curve as learnt in calculus.
 A: Your confusion comes from the fact that there are two types of line integrals. In calculus, you learnt that :

*

*You can integrate a scalar field $f : U\subset\mathbb{R}^n \rightarrow \mathbb{R}$ along a path $\gamma \subset U$ against the (curvilinear) length measure $\mathrm{d}s$, i.e. $\int_\gamma f(\vec{x})\mathrm{d}s$, where $\mathrm{d}s = \sqrt{\mathrm{d}x_1^2+\ldots+\mathrm{d}x_n^2}$. (Re)parametrizing the curve $\gamma$ by $\vec{x} : D\subset\mathbb{R} \rightarrow \gamma, t \mapsto \vec{x}(t)$, leads to $\int_Df(\vec{x}(t))\frac{\mathrm{d}s}{\mathrm{d}t}\mathrm{d}t = \int_Df(\vec{x}(t))|\dot{\vec{x}}(t)|\mathrm{d}t$.


*You can integrate a vector field $\vec{F} : U\subset\mathbb{R}^n \rightarrow \mathbb{R}^n$ along a path $\gamma \subset U$ against the vectorial measure $\mathrm{d}\vec{x}$, i.e. $\int_\gamma \vec{F}(\vec{x})\cdot\mathrm{d}\vec{x}$, where $\mathrm{d}\vec{x} = (\mathrm{d}x_1,\ldots,\mathrm{d}x_n)$. Reparametrizing the curve $\gamma$ by $\vec{x} : D\subset\mathbb{R} \rightarrow \gamma, t \mapsto \vec{x}(t)$, leads to $\int_D\vec{F}(\vec{x}(t))\cdot\frac{\mathrm{d}\vec{x}}{\mathrm{d}t}\mathrm{d}t = \int_D\vec{F}(\vec{x}(t))\cdot\dot{\vec{x}}(t)\mathrm{d}t$.
When $n=1$, there is no difference between scalar and vectorial fields; however, the first type of line integral becomes $\int_\gamma f(x)\mathrm{d}s$, with $\mathrm{d}s = |\mathrm{d}x|$, which is in fact never considered, while the second type of line integral coincides with the standard integral $\int_\gamma f(x)\mathrm{d}x$ $-$ the advantage of the second formula over the first one is the preservation of orientation.
All these considerations are still valid in $\mathbb{C}^n$, a fortiori in $\mathbb{C}$, where the measure $\mathrm{d}z \neq |\mathrm{d}z|$ corresponds to the second type of line integral, hence the change of variable $\int_\gamma f(z)\mathrm{d}z = \int_\gamma f(z(t))\dot{z}(t)\mathrm{d}t$, without the absolute value.
