Evaluate the following line integrals in the complex plane...

Evaluate the following line integrals in the complex plane by direct integration, that is, as in Chapter $6,$ Section $8,$ not using theorems from this chapter. (If you see that a theorem applies, use it to check your result.)
$\int_{1}^{i} z d z$ along the indicated paths:

Key Concepts

Direct Integration Technique

Direct integration refers to calculating the integral by substituting the parameterized curve directly into the integrand without relying on higher-level theorems. This involves differentiating the parametrization to find dz, substituting into the integral, and then integrating with respect to the parameter.

Antiderivatives and the Fundamental Theorem in Complex Analysis

While the exercise emphasizes direct integration, it is important to note that if the integrated function is analytic (holomorphic), the existence of an antiderivative allows one to evaluate the integral using the Fundamental Theorem of Calculus in complex analysis. This connection provides a check for the results obtained via direct integration.

Complex Line Integrals

A complex line integral involves integrating a complex-valued function along a curve in the complex plane. The integral is defined by parametrizing the curve, substituting the parametrization into the function, and integrating with respect to the parameter. This process is analogous to real line integrals but takes into account the behavior of complex functions over complex paths.

Path Parameterization

Parameterizing a path is the process of expressing the curve in the complex plane with a function z(t) where t varies over an interval. This is crucial for evaluating complex line integrals directly, as it converts the integral into a standard one-variable integral by replacing z and dz with z(t) and z?(t) dt, respectively.

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