Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{3}{x^{2}}-\frac{2}{x^{3}} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = -\frac{3}{x} + \frac{1}{x^2} + C \).

Step 1: Rewrite the function

Start by rewriting the given function in a more convenient form for integration using the power rule. The function is given as \( f(x) = \frac{3}{x^2} - \frac{2}{x^3} \). This can be rewritten as \( f(x) = 3x^{-2} - 2x^{-3} \).

Step 2: Apply the power rule for integration

The power rule for integration states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) for \( n eq -1 \). Apply this rule to each term separately:For \( 3x^{-2} \): \[ \int 3x^{-2} \, dx = 3 \cdot \frac{x^{-1}}{-1} = -3x^{-1} \]For \( -2x^{-3} \):\[ \int -2x^{-3} \, dx = -2 \cdot \frac{x^{-2}}{-2} = x^{-2} \]

Step 3: Combine the antiderivatives and add the constant of integration

Combine the results from the previous step. The antiderivative of \( f(x) \) is:\[ F(x) = -3x^{-1} + x^{-2} + C \]where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental technique in calculus essential for finding antiderivatives, or integrals, of functions. When you have a function in the form of \( x^n \), this rule simplifies the integration process. Here's how it works in a simple way: to integrate \( x^n \), increase the exponent by 1, making it \( n+1 \), and divide by this new power, \( \frac{x^{n+1}}{n+1} \). This rule is incredibly handy except when the exponent is -1, since it leads to a different result, the natural logarithm function.

This technique is applied to each term in a polynomial separately, making it methodical and straightforward. For example, consider \( 3x^{-2} \) as a part of our function. Using the Power Rule, add 1 to \(-2\) resulting in \(-1\), and then divide by \(-1\), yielding \(-3x^{-1}\). Apply the same process to \(-2x^{-3}\) and you get \(x^{-2}\). Understanding and mastering the Power Rule for Integration is crucial for solving a wide range of calculus problems, as it forms the basis for more complex integration techniques.

Integration Techniques

Integration can be approached with various techniques that cater to different types of functions. One of the simplest techniques is the Power Rule, as previously discussed. However, as problems become more complex, other techniques may be used, such as substitution, integration by parts, and partial fraction decomposition. Each method is tailored to handle specific function types more easily.

For the given problem, we rely heavily on the Power Rule. Yet, it's important to transform the function into a more workable form first. This involves rewriting the terms with negative exponents, like changing \(\frac{3}{x^2}\) to \(3x^{-2}\). This step is crucial for simplifying the integration process.

If the problem included products, quotients, or more complex expressions, substitution or integration by parts might come into play. While the Power Rule is fundamental, a robust understanding of all integration techniques is necessary to tackle diverse calculus problems effectively.

Calculus Problem Solving

Calculus problem solving often involves multiple steps and requires a methodical approach to ensure accuracy. Begin by carefully reading the problem to understand what is being asked. In this case, we aim to find the general antiderivative of a given function. To achieve this, it's essential to rewrite the expression, making it more suitable for integration.

After rewriting, apply the suitable integration technique. Here, it's the Power Rule. Each term of the rewritten function \( f(x) = 3x^{-2} - 2x^{-3} \) is integrated individually. It's crucial to work through each part sequentially and remember to combine results at the end, allowing for the constant of integration \( C \), which reflects the indefinite nature of the integral.

Ensuring clarity and consistency in each step is vital. Double-check each calculation to follow the correct arithmetic operations, especially when handling negative exponents and constants. A systematic approach helps avoid common pitfalls and leads to successful solutions in calculus problem solving.