Problem 24 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{1}{5}-\frac{2}{x^{3}}+2 x\right) d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{1}{5}x + \frac{1}{x^2} + x^2 + C \).

Step 1: Understand the Problem

The given problem is to find the indefinite integral of the function \( \int \left( \frac{1}{5} - \frac{2}{x^{3}} + 2x \right) \, dx \). This means we need to find a function whose derivative gives us \( \frac{1}{5} - \frac{2}{x^{3}} + 2x \).

Step 2: Integrate Each Term Separately

The expression inside the integral can be broken down into simpler parts: \( \frac{1}{5} \), \( -\frac{2}{x^3} \), and \( 2x \). 1. \( \int \frac{1}{5} \, dx = \frac{1}{5}x \).2. \( \int -\frac{2}{x^3} \, dx = \int -2x^{-3} \, dx = -2 \left( \frac{x^{-3+1}}{-3+1} \right) = \frac{1}{x^2} \).3. \( \int 2x \, dx = 2 \left( \frac{x^{2}}{2} \right) = x^2 \).

Step 3: Combine the Integrals

Combine the results from Step 2 to get the general antiderivative:\[\frac{1}{5}x + \frac{1}{x^2} + x^2 + C\]where \( C \) is the constant of integration.

Step 4: Check by Differentiation

Differentiate the result from Step 3 to ensure it matches the original function:1. The derivative of \( \frac{1}{5}x \) is \( \frac{1}{5} \).2. The derivative of \( \frac{1}{x^2} \) is \( -\frac{2}{x^3} \).3. The derivative of \( x^2 \) is \( 2x \).4. The derivative of the constant \( C \) is 0.Thus, the derivative \( \frac{1}{5} - \frac{2}{x^3} + 2x \) matches the original function inside the integral.

Key Concepts

Antiderivative

Understanding antiderivatives, also known as indefinite integrals, is crucial for working with integrals. An antiderivative is a function that reverses differentiation. In other words, if you have an antiderivative, taking its derivative will return you to your original function. For the exercise at hand, our task was to find the antiderivative of \[\frac{1}{5} - \frac{2}{x^3} + 2x\].

It's important to recognize that the basic concept involves finding a function whose derivative equals the given function, considering each component individually. This helps break down more complex functions into manageable parts. Remember, every function can have infinitely many antiderivatives. That's why we add a constant of integration, denoted as \( C \). This constant represents all the vertical shifts of our antiderivative on a graph and is crucial for obtaining the most 'general' antiderivative.

Differentiation

Differentiation is the process of finding the derivative of a function. It's a core concept in calculus, focusing on determining the rate of change or the slope of a function at any point. In our exercise, it was used to verify the accuracy of our antiderivative.

We started with our calculated antiderivative:\[\frac{1}{5}x + \frac{1}{x^2} + x^2 + C\] and then differentiated it.

• Deriving \(\frac{1}{5}x\) provides \(\frac{1}{5}\).
• Deriving \(\frac{1}{x^2}\) returns \(-\frac{2}{x^3}\).
• Deriving \(x^2\) gives \(2x\).
• The derivative of a constant like \(C\) is 0.

Each derivative coincides perfectly with our original function inside the integral. Verifying with differentiation ensures that we've correctly identified the antiderivative. Always double-check your results, as it's a surefire way to catch any missteps.

Integration Rules

Integration rules simplify the process of finding antiderivatives. These rules are similar to differentiation rules and help to quickly integrate various types of expressions. For example, in our given integral, the function was split into each of its terms: \(\frac{1}{5}\), \(-\frac{2}{x^3}\), and \(2x\).

Here are a few key rules used:

• The constant rule, which states that the integral of a constant \(a\) is \(ax\).
• The power rule for integration, a counterpart to the power rule for differentiation, which handles expressions of the form \(x^n\) and results in \(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\).
• The sum rule, which allows you to integrate a function term by term.

Each term in the original function was treated individually using these rules, then recombined to find the most general antiderivative. Being familiar with these rules will significantly expedite the process of solving integrals.