Problem 8 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=1-\frac{1}{x}+\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The general antiderivative is: \\( ext{Integral} = x - \ln|x| - x^{-1} + C\\), where \\(C\\) is a constant.

Step 1: Identify the Structure of the Function

The given function is composed of three terms: a constant term, a term that involves \(x^{-1}\), and a term that involves \(x^{-2}\). We can write it as \(f(x) = 1 - rac{1}{x} + rac{1}{x^2}\), which is equivalent to \(f(x) = 1 - x^{-1} + x^{-2}\).

Step 2: Apply the Power Rule for Antiderivatives

To find the antiderivative of a polynomial in the form \(x^n\), we use the formula \(rac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. Apply this rule to each term in the function separately.

Step 3: Antidifferentiate Each Term

Antidifferentiate the constant term \(1\) to get \(x + C\). For the term \(x^{-1}\), the antiderivative is \(-\ln|x| + C\). For the term \(x^{-2}\), use the power rule to get \(-x^{-1} + C\).

Step 4: Combine the Antiderivatives

Combine the antiderivatives from Step 3, remembering to include one constant of integration: \( ext{Integral} = x - \ln|x| - x^{-1} + C\), where \(C\) is the constant of integration.

Key Concepts

Antiderivative

An antiderivative, in calculus, refers to a function that reverses the process of differentiation. When you have a function, such as the one we are examining here, finding its antiderivative means determining a function whose derivative would yield the original function. This is often called "indefinite integration," and it helps understand the accumulation of quantities. It’s basically the opposite of taking a derivative. Just like you might remember multiplication as the opposite of division, antiderivatives provide us with a tool to "undo" differentiation.

Power Rule for Antiderivatives

The power rule for antiderivatives is a fundamental technique for integrating functions with powers of variables. If you have a basic polynomial function like \(x^n\), the power rule helps you find an antiderivative using a simple formula.

• For any function \(x^n\), where \(neq -1\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\).

In our specific problem:

• The term \(x^{-2}\) becomes \(-x^{-1} + C\).

Remember, we cannot directly use the power rule for \(x^{-1}\) because it behaves differently. Instead, its antiderivative is \(-\ln|x| + C\). This highlights how crucial understanding the formula thoroughly is, as exceptions do occur.

Constant of Integration

When dealing with indefinite integrals, or antiderivatives, a constant of integration (denoted as \(C\)) is always added. This constant represents an infinite set of possible antiderivatives. Why is that? Because the derivative of any constant is zero, adding \(C\) allows for all variations of a function which could have produced the same original derivative.In practical terms, when finding the antiderivative of any function, by always adding \(C\), we're acknowledging that there isn't one unique antiderivative, but a whole family of them, differing by a constant.

Functions of a Single Variable

In calculus, many problems deal with functions of a single variable, like the function in our example, which varies with respect to \(x\). Such functions are straightforward: they depend on one independent variable allowing for direct and simplified calculus operations like finding derivatives or antiderivatives.By focusing on one variable, it becomes easier to apply calculus rules, as each term transformation is directly associated with changing \(x\). In this problem, the terms were respectively \(1, x^{-1},\) and \(x^{-2}\), each being individually manageable due to their nature as simple expressions in \(x\). This aids in practicing fundamental calculus techniques like using power rules effectively, as transformations affect only one variable’s power, making calculations more intuitive and less error-prone.