Problem 38 - Applied Calculus

Question

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(t+2)(t-4)}{t^{2}} d t\)

Step-by-Step Solution

Answer

The indefinite integral is \\( t - 2 \ln |t| + \frac{8}{t} + C \\\).

Step 1: Expand the Numerator

First, expand the expression in the numerator. Multiply \( (t+2)(t-4) \) to get \( t^2 - 4t + 2t - 8 \), which simplifies to \( t^2 - 2t - 8 \).

Step 2: Separate the Fraction

Rewrite the integral by separating the terms in the fraction: \(\int \frac{t^2 - 2t - 8}{t^2} \, dt = \int \left( \frac{t^2}{t^2} - \frac{2t}{t^2} - \frac{8}{t^2} \right) \, dt. \)

Step 3: Simplify Each Term

Simplify each term of the fraction: \(\int \left( 1 - \frac{2}{t} - \frac{8}{t^2} \right) \, dt. \)

Step 4: Integrate Term by Term

Integrate each term separately: - \( \int 1 \, dt = t \) - \( \int -\frac{2}{t} \, dt = -2 \ln |t| \) - \( \int -\frac{8}{t^2} \, dt = 8\frac{1}{t} = 8t^{-1} \)

Step 5: Combine the Integrated Terms

Combine the results of the integration: \(t - 2 \ln |t| + \frac{8}{t} + C, \)where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Understanding integration techniques is vital when solving indefinite integrals. An indefinite integral essentially represents a family of functions whose derivative is the integrand. In this exercise, we use basic techniques to find the integral of a rational function.

The function given by \[\int \frac{(t+2)(t-4)}{t^{2}} \ dt\]is not immediately integrable. Therefore, we use algebraic manipulation, such as expanding and simplifying, to rewrite it in a more manageable form. This step is crucial for applying basic integration rules. Integration by parts or substitution might be necessary if the problem becomes more complex, but for this particular exercise, the integration can be performed directly after simplifying the expression.

Breaking down complex expressions and recognizing patterns helps in selecting the appropriate integration technique. Familiarity with basic integration formulas, like \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]when \( n eq -1 \), is essential to perform integration effectively.

Algebraic Manipulation

Algebraic manipulation is a critical part of simplifying expressions before integrating. In this exercise, the hint suggests using algebra. Let's break down why it is important:

• **Expansion**: The product \((t+2)(t-4)\) needs expansion. Multiply these polynomials to get a quadratic expression \( t^2 - 2t - 8 \).
• **Fraction Separation**: After expansion, the expression is divided by \(t^2\). Separate this into individual fractions: \[\frac{t^2}{t^2} - \frac{2t}{t^2} - \frac{8}{t^2}\]
• **Simplification**: Simplify each term to make integration straightforward:
  • \(\frac{t^2}{t^2} = 1\)
  • \(\frac{2t}{t^2} = \frac{2}{t}\)
  • \(\frac{8}{t^2} = 8t^{-2}\)

After simplification, the expression becomes easy to integrate term by term, avoiding more complex techniques that are unnecessary here.

Calculus Step-by-Step Solutions

A step-by-step solution is key when handling calculus problems. It ensures clarity and understanding of each phase of problem-solving. Let’s consider the steps for solving the integral:

1. **Expansion and Fraction Separation**: Initially, transform the integrand into a simpler form that reveals the structure of the integral. This was done via expansion and separating into individual terms.

2. **Integrate Each Term**: With the expression simplified, each term can be integrated separately:
- Integrating \(1\) with respect to \(t\) is straightforward, yielding \(t\).- The term \(-\frac{2}{t}\) is integrated using the natural logarithm rule, yielding \(-2 \ln |t|\).- Lastly, \(-\frac{8}{t^2}\) simplifies to a power rule, integrating to \(-8t^{-1} = \frac{8}{t}\).
3. **Combine and Interpret**: The final step is to combine all individually integrated components with the constant of integration \(C\), representing all potential solution functions.Each step in calculus builds upon the last. By methodically applying integration techniques and algebraic manipulation, the final solution, \[t - 2 \ln |t| + \frac{8}{t} + C\]becomes clear and comprehensible.