Problem 25 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(x^{2}+x+1+x^{-1}+x^{-2}\right) d x\)

Step-by-Step Solution

Answer

\( \frac{x^3}{3} + \frac{x^2}{2} + x + \ln |x| - \frac{1}{x} + C \)

Step 1: Identify and Separate Terms

Separate the integrand into individual terms: \[ x^2, \quad x, \quad 1, \quad x^{-1}, \quad x^{-2} \] This allows us to integrate each term separately.

Step 2: Apply the Power Rule

We use the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for each term:1. For \( x^2 \), the integral is \( \frac{x^{3}}{3} \).2. For \( x^1 \), the integral is \( \frac{x^{2}}{2} \).3. For \( x^0 \) (or 1), the integral is \( x \).4. For \( x^{-1} \), recall \( \int x^{-1} \, dx = \ln |x| \).5. For \( x^{-2} \), the integral is \( \frac{x^{-1}}{-1} = -x^{-1} \).

Step 3: Combine the Integrals

Combine all the integrals calculated in the previous step:\[ \frac{x^3}{3} + \frac{x^2}{2} + x + \ln |x| - x^{-1} + C \]

Step 4: Simplify the Expression

Write the final expression neatly:\[ \int \left( x^2 + x + 1 + x^{-1} + x^{-2} \right) \, dx = \frac{x^3}{3} + \frac{x^2}{2} + x + \ln |x| - \frac{1}{x} + C \]

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept used in calculus to find the antiderivative or indefinite integral of polynomial expressions. When you are given an expression in the form of \(x^n\), you can integrate it using the formula:

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

Here, \(n\) is any real number except \(-1\), because integrating \(x^{-1}\) requires a different approach involving the natural logarithm. This formula helps transform a differentiable function into an antiderivative by increasing the exponent by 1, then dividing by this new exponent. For example, to integrate \(x^2\), you increase the power to 3, resulting in \(\frac{x^3}{3}\) as part of the solution. This rule simplifies finding the antiderivative and turns integration into a straightforward process when dealing with polynomial terms.

Integration Techniques

Integration techniques involve various methods used to find the indefinite integral or the antiderivative of functions. For straightforward polynomial terms, using the power rule often suffices, but there are instances where you need additional strategies. Here are some common techniques:

• Polynomials: Using the power rule, each term of a polynomial can be integrated separately and then summed to get the final result.
• Logarithmic integration: When integrating \(x^{-1}\), the result is \(\ln |x|\) due to a special case different from the power rule.

When dealing with the original exercise, each term of the polynomial expressed can be tackled individually. For example, separating the terms like \(x^2, x,\) and \(x^{-2}\) allows you to treat each one separately with appropriate techniques such as the power rule or other special cases like logarithms for \(x^{-1}\). Being able to recognize and apply these techniques effectively is crucial for efficiently solving integration problems.

Integration Constants

When calculating indefinite integrals, it's crucial to add a constant, commonly denoted as \(C\), to the result. This integration constant arises because when you differentiate a constant, it becomes zero, meaning there are infinitely many antiderivatives that differ by a constant.

• Why is the constant added? Since the derivative of a constant is zero, any function \(f(x) + C\) has the same derivative as \(f(x)\), and thus, without the constant, you miss other possible solutions.
• The role of \(C\): It signifies all possible original functions that could lead to the same derivative function.

In the context of the exercise, after integrating each term of the expression \(x^2 + x + 1 + x^{-1} + x^{-2}\), the final integral is expressed as: \(\frac{x^3}{3} + \frac{x^2}{2} + x + \ln |x| - \frac{1}{x} + C\). Ensure to always include \(C\) in indefinite integrals to represent the most general form of the antiderivative.