Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(x^{2}+x+x^{-3}\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: \(F(x) = \frac{x^3}{3} + \frac{x^2}{2} - \frac{1}{2x^2} + C\).

Step 1: Identify the terms for integration

The given function can be written as \(f(x) = x^2 + x + x^{-3} \) We will find the indefinite integral, \(F(x)\), for each term separately, using the power rule for integration: $$\int\left(x^{2}+x+x^{-3}\right) d x = \int x^2\, dx + \int x\, dx + \int x^{-3}\, dx$$

Step 2: Use the power rule for integration on each term

The power rule for integration states that: $$\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$$ Considering each term individually: 1. For the term \(x^2\): $$\int x^2\, dx = \frac{x^{2+1}}{2+1} + C_1 = \frac{x^3}{3} + C_1$$ 2. For the term \(x\): $$\int x\, dx = \frac{x^{1+1}}{1+1} + C_2 = \frac{x^2}{2} + C_2$$ 3. For the term \(x^{-3}\): $$\int x^{-3}\, dx = \frac{x^{-3+1}}{-3+1} + C_3 = \frac{x^{-2}}{-2} + C_3$$

Step 3: Add the results from the terms and the constants

Now, sum the results of the integration from each term and combine the constants: $$F(x) = \left(\frac{x^3}{3} + C_1\right) + \left(\frac{x^2}{2} + C_2\right) + \left(\frac{x^{-2}}{-2} + C_3\right)$$ We can combine the constants into one, say 'C', by writing \(C_1 + C_2 + C_3 = C\). Thus, the antiderivative of the given function is: $$F(x) = \frac{x^3}{3} + \frac{x^2}{2} - \frac{1}{2x^2} + C$$

Key Concepts

Power Rule for Integration

When tackling indefinite integrals, one of the most crucial techniques is the power rule for integration. This rule is a straightforward yet powerful tool for finding antiderivatives of polynomial functions. If you have a term in the form of \(x^n\), the power rule helps us integrate it efficiently.

Here's the formula for applying the power rule:

• \(\int x^n\, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\).

This rule allows you to find the antiderivative by increasing the exponent by 1 and dividing by the new exponent. Remember, if \(n\) is \(-1\), this rule does not apply; instead, \(\int x^{-1} dk = \ln| x| + C\).
In our example, the power rule was key to integrating each term, such as \(x^2\), \(x\), and \(x^{-3}\). By applying the rule individually to these terms, you can smoothly reach the integral of the whole function.

Integration Techniques

Integration techniques simplify the process of finding the antiderivative of a function. The core idea is to manage complex expressions by breaking them down into simpler parts or using specific formulas.

For functions made up of several terms like our example \(x^2 + x + x^{-3}\), splitting into individual terms makes integration more manageable. This divide-and-conquer approach is crucial when applying the power rule or other integration methods.
Additionally, recognizing patterns or transformations can be handy. While the example given is simple, in other cases, you might encounter more advanced techniques such as substitution or integration by parts, particularly if the power rule does not apply directly. A solid foundation in these basic techniques supports tackling more complex problems later on.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative returns the original function. It's a fundamental concept that bridges differentiation and integration.

In our context, seeking an antiderivative involves finding a function \(F(x)\) such that \(F'(x) = f(x)\). The solution presents this as:

• \(F(x) = \frac{x^3}{3} + \frac{x^2}{2} - \frac{1}{2x^2} + C\)

Here, \(C\) is a constant of integration. When calculating indefinite integrals, this constant comes into play because any constant differentiates to zero.
Thus, the concept of an antiderivative highlights the infinite set of solutions that share the same derivative, emphasizing the importance of the constant \(C\) in providing the complete solution.