91Ó°ÊÓ

The power rule for integration is a fundamental concept in calculus that greatly simplifies the process of finding antiderivatives. It states that if you have a function of the form \( z^n \), where \( n eq -1 \), its integral is given by:\[\int z^n \, dz = \frac{z^{n+1}}{n+1} + C,\]where \( C \) represents the constant of integration. This rule allows us to handle a wide variety of polynomial functions easily.To apply the power rule:

• Identify the exponent \( n \) of the term.
• Add 1 to \( n \) to get \( n+1 \).
• Divide the term \( z^{n+1} \) by \( n+1 \).

This method streamlines integration significantly, making it much simpler to find the integral of powers of \( z \). Note, however, that this rule does not apply when \( n = -1 \), a scenario that requires a different approach involving logarithms.

A definite integral is a type of integral that calculates the area under a curve between two specified limits. Unlike indefinite integrals, which include a constant \( C \), definite integrals result in a specific numerical value. The format for a definite integral is:\[\int_{a}^{b} f(z) \, dz,\]where \( a \) and \( b \) are the lower and upper limits of integration, respectively.To evaluate a definite integral:

• Find the antiderivative of the function \( f(z) \).
• Substitute the upper limit \( b \) into the antiderivative to find \( F(b) \).
• Substitute the lower limit \( a \) into the antiderivative to find \( F(a) \).
• Subtract \( F(a) \) from \( F(b) \) to get the final area value.

This process essentially calculates the accumulation of the quantity represented by \( f(z) \) over the interval from \( a \) to \( b \). It moves beyond just finding slopes or rates, allowing us to determine the total change between two points.

When dealing with integrals of negative exponents, first transform the terms to a form where the power rule can be easily applied. For example, for terms like \( \frac{1}{z^n} \), write them as \( z^{-n} \). This makes the integration process straightforward since you will then use the power rule:\[\int z^{-n} \, dz = \frac{z^{-n+1}}{-n+1} + C.\]Here's how to integrate each term:

• Convert division expressions into negative exponents.
• Apply the power rule for integration to each term.
• If necessary, simplify the expression to return to positive exponents or fractions.

As seen in the original exercise, terms such as \( \frac{1}{z^3} \) and \( \frac{3}{z^2} \) can be rewritten as \( z^{-3} \) and \(-3z^{-2} \), respectively. After applying the power rule, transform back to fractions if required, to present the results in a more familiar form. This conversion is a vital step in evaluating integrals involving negative exponents.