91Ó°ÊÓ

The power rule for integration is a fundamental concept of calculus. It's especially handy when you need to find the integral of expressions in the form of a power function. The rule states that the integral of a function \(x^n\) is given by \(\int x^n \, dx = \frac{1}{n+1} x^{n+1} + C\), where \(C\) is the constant of integration (more on that later). This rule only applies when \(n \eq -1\).

In the problem, we see an expression like \( (z-4)^{-5} \). Even though it's not just a plain power of \(z\), we can still use the power rule by treating \(z-4\) as the base variable just like \(x\). The exponent \(-5\) is our \(n\) in this scenario.

By applying the power rule, we add \(1\) to the exponent and then divide by this new exponent. It's crucial to adjust the constant outside the integral as it forms a part of the final answer.

When finding an indefinite integral, you will always come across a term called the constant of integration, denoted by \(C\). Why is it there? The reason is simple; differentiation and integration are inverse operations. And during the process of differentiation, any constant term disappears.

Hence, when you integrate a function, you have to account for all possible functions you could have started with. That's why we add this constant term \(C\). It's there to reflect that our solution is part of a family of functions, and any value of \(C\) describes a different member of that family.

So, in our solution, after we find the integral using the power rule, the result includes this \(+ C\). It's an important part of the final answer and should not be overlooked.

Sometimes in integration, a direct approach isn't enough. Here comes the idea of changing variables, which is also commonly known as "substitution." This is a technique used to simplify an integral, making it easier to solve.

In the given problem, we treat \(z-4\) as the variable. This is a form of substitution itself by recognizing that \(z-4\) is easier to manage than something more complex. This approach helps simplify the integral, allowing the power rule to be easily applied.

• Identify the expression that is complicated, which in this case is \(z-4\).
• Consider substituting it with a new variable, though in this problem it is handled as is.
• After integrating with the simpler expression, maintain the overall constant to obtain the clear solution.

The change of variables is not an overkill but rather an essential tool, especially when straightforward methods don't work. It's like having a new perspective on a problem that helps in breaking it down into solvable pieces.