Integrate by the first method or state why it does not apply...

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.
$\int_{c} \operatorname{Re} z d e, C$ the shortest path fiom 0 to $1+i$

Key Concepts

Complex Line Integrals

Complex line integrals involve integrating a complex-valued function along a specified path in the complex plane. This process is analogous to real line integrals but takes into account the complex nature of both the function and the differential element. Understanding the definition and properties of these integrals is essential, as the result may depend on the chosen path, especially when dealing with non-holomorphic functions.

Parameterization of Curves

Parameterizing a curve in the complex plane means expressing the complex variable as a function of a real parameter, typically t, over a given interval. This method allows one to transform a complex line integral into an integral with respect to a real variable, making it easier to compute. It is particularly useful when the function being integrated does not have an easily obtainable antiderivative in the complex sense.

Analyticity and Holomorphic Functions

A function is said to be holomorphic if it is complex differentiable at every point in an open region. Holomorphic functions have many useful properties, one of which is that their line integrals between two points are independent of the path taken. When a function is not holomorphic, as is the case with functions involving the real part of z, the standard antiderivative method may not apply, necessitating alternative approaches such as direct parameterization.

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