Problem 39 - Applied Calculus

Question

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(x-2)^{3}}{x} d x\)

Step-by-Step Solution

Answer

The integral is \( \frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| + C \).

Step 1: Simplify the Integrand

First, perform algebraic division to simplify the integrand. Divide \((x-2)^3\) by \(x\):\[(x-2)^3 = (x^3 - 6x^2 + 12x - 8)\].By dividing each term by \(x\), we get: \[\frac{x^3}{x} - \frac{6x^2}{x} + \frac{12x}{x} - \frac{8}{x} = x^2 - 6x + 12 - \frac{8}{x}\].The integral becomes:\[\int (x^2 - 6x + 12 - \frac{8}{x}) \, dx\].

Step 2: Integrate Term by Term

Now integrate each term separately:\[\int x^2 \, dx = \frac{x^3}{3}\], \[\int (-6x) \, dx = -3x^2\], \[\int 12 \, dx = 12x\], \[\int (-\frac{8}{x}) \, dx = -8\ln|x|\].

Step 3: Combine the Integrals

Combine the results of each individual integral:\[\frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| + C\],where \(C\) is the constant of integration.

Key Concepts

Algebraic Simplification

Before diving into integrating complex expressions, simplifying the integrand through algebraic manipulation can make the solution much easier. To handle an expression such as \( \frac{(x-2)^3}{x} \), it's useful to simplify by expanding the numerator fully. This way, each term can be divided by the denominator individually.For example:

• Expand \((x-2)^3\) to get \(x^3 - 6x^2 + 12x - 8\).
• Divide each term by \(x\), resulting in \(x^2 - 6x + 12 - \frac{8}{x}\).

This simplification turns a seemingly complex fraction into simpler polynomial terms, thereby paving the path for straightforward integration. By taking the time to simplify algebraically, it becomes easier to apply standard integration rules.

Integration Techniques

Once the integrand has been simplified, the next step is to integrate it term by term. Each part of the polynomial can be tackled using basic integration rules.Here are some useful techniques:

• For a power of \(x\), such as \(x^n\), use the formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\), applicable for non-zero \(n\).
• For constants multiplied by \(x\), such as \(-6x\), simply apply \(\int -6x \, dx = -3x^2\).
• For constant terms \(c\), the integration is straightforward: \(\int 12 \, dx = 12x\).
• For the term \(-\frac{8}{x}\), recognize it as a natural logarithmic function \(\int -\frac{8}{x} \, dx = -8\ln|x|\).

These techniques simplify integration into manageable steps, helping to systematically work through different types of terms in the expression.

Calculus Problem Solving

In calculus problem solving, understanding each step in the integration process is crucial for finding the final solution and verifying its correctness. After integrating each term in the integrand separately, it's essential to combine them properly to form the complete solution.Steps to finalize your solution:

• Integrate each term separately: Start from the highest power of \(x\) and proceed stepwise to easier terms, including constants.
• Combine the integrated terms: In our example, this yields: \( \frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| \).
• Don't forget the constant of integration \(C\): This is a crucial part when dealing with indefinite integrals as it represents the family of antiderivatives.

By following these steps, students develop a structured approach to calculus problems, which helps in achieving the correct solution efficiently. Each solved integration problem strengthens problem-solving skills and comprehension of calculus concepts.