Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{x^{3}-8 x}{2 x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \int \frac{x^3 - 8x}{2x^2} dx \) is \( \frac{1}{4} x^2 - 4 \ln|x| + C \).

Step 1: Simplifying the Integral

The first step to solve this problem involves simplifying the integral. The given integral expression can be simplified into simpler terms by splitting the fractions: \( \int (\frac{x^3}{2x^2} - \frac{8x}{2x^2}) dx \). Further simplification yields \( \int (\frac{1}{2}x - \frac{4}{x}) dx \).

Step 2: Integrating the Simplified Expression

In the simplified expression, the integral can be computed part by part. For \(\frac{1}{2}x\), applying the power rule for integrals \(\int x^n dx = \frac{1}{n+1}x^{n+1} + C\), we get \( \frac{1}{2} \int x dx = \frac{1}{4} x^2 \). For \(\frac{4}{x}\), applying the formula \(\int \frac{1}{x} dx = \ln|x| + C\), we get \( - 4 \int \frac{1}{x} dx = -4 \ln|x| \)

Step 3: Writing the Final Answer

Combine the two parts to give the final integral function: \( \frac{1}{4} x^2 - 4 \ln|x| + C \). Here \( C \) is the constant of integration.

Key Concepts

Integration Formulas

Integration is a fundamental concept in calculus that allows us to find areas under curves or solve various problems involving the accumulation of quantities. One of the key concepts is knowing which integration formula to use. In this exercise, two important integration formulas are applied: the Power Rule and Logarithmic Integration. These help us break down complex expressions into manageable parts.

• The Power Rule: Useful when dealing with terms like \( x^n \).
• Logarithmic Integration: Necessary when integrating functions like \( \frac{1}{x} \).

Identifying the correct formula simplifies the integration process, making it more straightforward and manageable. Effective use of these formulas is essential for solving integrals correctly.

Power Rule for Integration

The Power Rule for Integration is a simple yet powerful tool in calculus. It's similar to the Power Rule in differentiation but focuses on finding antiderivatives. It's used whenever you're integrating a term of the form \( x^n \).
The rule states: If \( n eq -1 \), then \( \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \). Here, \( C \) represents the constant of integration, capturing the indefinite aspect of the integral.
When applying the Power Rule, consider the index of \( x \) in the expression. For the term \( \frac{1}{2} x \), we have \( n = 1 \). Applying the Power Rule gives us \( \frac{1}{4} x^2 \).
This step showcases the rule's simplicity and its application in breaking down the integration into digestible pieces.

Logarithmic Integration

Logarithmic integration is essential when dealing with expressions where the variable is in the denominator, such as \( \frac{1}{x} \). This particular case is special because it doesn't fit the Power Rule's typical form, requiring a unique technique.
The integral \( \int \frac{1}{x} \, dx = \ln |x| + C \) defines this rule. It highlights the natural logarithm as the antiderivative for \( \frac{1}{x} \). In our exercise, the term \( -\frac{4}{x} \) requires this formula. So, applying logarithmic integration, we get \( -4 \ln |x| \).
Logarithmic integration is often encountered in situations dealing with rates of change or growth, where the variable's influence is inversely proportional. Mastering this approach ensures you can tackle a broad range of integrals efficiently.