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U-substitution is a pivotal technique in integral calculus that helps simplify complex integrals, making them easier to solve. This technique involves substituting part of the integral with a new variable, usually denoted as \( u \).

In the context of our exercise, we chose \( u = 4 + z^2 \). This choice is strategic because it simplifies the integral by turning a complicated expression into a simpler one. By differentiating \( u \) with respect to \( z \), we find that \( du = 2z \, dz \), which implies \( z \, dz = \frac{1}{2} \, du \).

This conversion allows us to change the variable of integration from \( z \) to \( u \). So, the integral becomes easier to handle using the known integral tables. By using u-substitution, the integral becomes more approachable, paving the way to use integral tables or find the antiderivative more directly.

Integral tables are comprehensive lists that contain solutions to common integrals and can be extremely useful for solving complex problems quickly. In the exercise, after applying u-substitution, we are expected to refer to the integral table to find a direct solution.

The integral \( \int \frac{\sqrt{u}}{\sqrt{u-4}} \, du \) resembles forms that are typically listed in these tables. Once rewritten, our integral can be matched to a known form, allowing for immediate use of a pre-calculated antiderivative result.

Using these tables can save a significant amount of time, as they provide results for integrals that might otherwise require lengthy computation. They serve as a shortcut by eliminating the need to derive antiderivatives from scratch, which is particularly helpful in exams and challenging homework assignments.

Antiderivatives, also known as indefinite integrals, are the inverse operation of derivatives. When you integrate a function, you're essentially finding its antiderivative.

In our exercise, after using u-substitution and employing the integral table, the function's antiderivative is found quickly. The key step lies in recognizing or transforming the integral to a form whose antiderivative is known or accessible. In typical situations, once the antiderivative \( F(u) \) is found, we substitute back the original variable, obtaining the solution in terms of \( z \).

It's crucial to remember that constants such as \( C \) can emerge as part of any indefinite integral. Since differentiation of a constant is zero, they have no effect when derivatives are taken, but can affect the family of functions we represent. This constant ensures all possible antiderivatives are covered.