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U-substitution is a technique used in integration that allows us to simplify an integral by substituting a part of it with a single variable, usually labeled as \( u \). This method is particularly helpful when dealing with composite functions or when a straightforward integration is challenging.
To apply u-substitution, follow these steps:

• Identify a portion of the integral that can be replaced with \( u \). In this exercise, \( z+1 \) is chosen as \( u \). The derivative of \( u \) with respect to \( z \) gives \( du = dz \), which is perfect for substitution.
• Express the rest of the integrand in terms of \( u \). This usually involves solving for the original variable, \( z = u - 1 \).
• Rewrite the integral using \( u \) and carry out the integration with respect to \( u \).

This substitution often simplifies complex expressions, making the integration process more straightforward and manageable.

Antiderivatives, sometimes called indefinite integrals, are the reverse process of derivatives. They help us find a function that, when differentiated, gives back the original function. To find an antiderivative, consider integrating the function term-by-term.
In this particular problem, the expression was expanded to \( u^{4/3} - u^{1/3} \). Each term needs to be integrated separately:

• For \( u^{4/3} \), compute the antiderivative by adding 1 to the exponent, then divide by the new exponent. The result is \( \frac{u^{7/3}}{7/3} = \frac{3}{7}u^{7/3} \).
• For \( -u^{1/3} \), similarly, the antiderivative is \( \frac{u^{4/3}}{4/3} = \frac{3}{4}u^{4/3} \).

Combining these antiderivatives provides us with a complete solution, which includes a constant of integration, \( C \), to represent any possible constant term that would have been lost during differentiation.

In the context of integrals, it's essential to distinguish between definite and indefinite integrals.
An **indefinite integral**, like the one in this exercise, represents a family of functions. It includes an arbitrary constant \( C \) because the antiderivative could technically shift up or down vertically. The result of integrating in this example is:\[\frac{3}{7}(z+1)^{7/3} - \frac{3}{4}(z+1)^{4/3} + C\]which is a general form and not tied to any specific limit.

• Indefinite integrals often appear in contexts where you're finding the general antiderivative without specific boundaries.

On the other hand, a **definite integral** provides a specific numerical value. It evaluates the area under the curve between two points, defined as the limits of integration. When working with definite integrals, there's no need to add the constant \( C \) since definite integrals yield numerical results arising from specific interval calculations.