91Ó°ÊÓ

When you're finding the indefinite integral of a function, it's crucial to understand the integrand—the expression inside the integral sign. Analyzing the integrand helps you determine the method for integration. In our exercise, the integrand is \( \frac{1}{\sqrt[3]{2}} \), which is simply a constant as it does not vary with the variable \( z \). Recognizing this constant allows us to apply specific rules easily, like those for a constant of integration. Since the integrand does not involve \( z \), you immediately know that the integral is straightforward. The rule for integrating a constant tells us there isn't any complicated math involved here, just identifying the constant and integrating it with respect to \( z \).

Whenever you calculate an indefinite integral, you add a constant of integration, often denoted as \( C \). This is because the process of integration is essentially the reverse of differentiation and usually, many functions share the same derivative, differing only by a constant. In essence, when you take a derivative, any constant vanishes, so when integrating, we add back this unknown constant to reflect all possible original functions.The constant of integration \( C \) ensures your solution includes all potential functions that could have led to the same derivative. For example, when we integrate \( c = \frac{1}{\sqrt[3]{2}} \), the result is \( \frac{1}{\sqrt[3]{2}} \cdot z + C \), where \( C \) stands for any constant that, when differentiated away, would yield the original constant integrand.

One of the most fundamental rules of integration is the integral of a constant. If the integrand is a constant, say \( c \), the integral is straightforward: \( \int c \, dz \) equals \( c \cdot z + C \), where \( C \) is the constant of integration.In our example, the constant is \( \frac{1}{\sqrt[3]{2}} \). Applying the rule of integrating a constant, we multiply this constant by the variable \( z \), yielding \( \frac{1}{\sqrt[3]{2}} \cdot z \). We add the constant of integration \( C \) to ensure the solution reflects all possible antiderivatives of the original constant term. This approach allows us to handle simple integrands with ease, making integration of constants a basic yet powerful tool in calculus.