Problem 37 - Applied Calculus

Question

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(t-1)(t+3)}{t^{2}} d t\)

Step-by-Step Solution

Answer

The indefinite integral is \(t + 2 \ln |t| + \frac{3}{t} + C.\)

Step 1: Expand the Numerator

To simplify the expression, we first expand the numerator. We have \((t-1)(t+3)\). Distribute \(t-1\) over \(t+3\), which gives: \[(t-1)(t+3) = t \cdot t + t \cdot 3 - 1 \cdot t - 1 \cdot 3 = t^2 + 3t - t - 3 = t^2 + 2t - 3.\]

Step 2: Rewrite the Integral

The integral is now: \[\int \frac{t^2 + 2t - 3}{t^2} \, dt.\] We rewrite it by separating the fraction: \[\int \left(\frac{t^2}{t^2} + \frac{2t}{t^2} - \frac{3}{t^2}\right) \, dt = \int (1 + \frac{2}{t} - \frac{3}{t^2}) \, dt.\]

Step 3: Integrate Term by Term

Now we integrate each term separately: 1. \(\int 1 \, dt = t\)2. \(\int \frac{2}{t} \, dt = 2 \ln |t|\)3. \(\int -\frac{3}{t^2} \, dt = \int -3t^{-2} \, dt = 3t^{-1} = \frac{3}{t}\)Thus, the integral becomes: \[t + 2 \ln |t| + \frac{3}{t} + C,\] where \(C\) is the constant of integration.

Key Concepts

Algebraic Manipulation

When dealing with integrals involving algebraic fractions, using algebraic manipulation can greatly simplify the process. This technique involves rewriting the expression into a form that is easier to integrate. Take, for instance, the expression

• \(\int \frac{(t-1)(t+3)}{t^{2}} dt\)

Here, the numerator is a product of two binomials. By expanding the expression using the distributive property, it becomes

• \((t-1)(t+3) = t^2 + 2t - 3\).

This expansion allows us to rewrite the integral into separate terms, each of which can be individually integrated. Using algebraic methods like expansion, factoring, or separating fractions helps in managing more complex expressions elegantly. It often unveils simpler terms that align well with basic integration rules.

Calculus Integration

Integration is the process of finding the integral of a function, and it can either be definite or indefinite. In indefinite integrals, we focus on finding the antiderivative of a function without specific limits. For the expression

• \(\int (1 + \frac{2}{t} - \frac{3}{t^2}) dt\),

we integrate each term individually:

• The integral of 1 with respect to \(t\) is simply \(t\).
• For the term \(\frac{2}{t}\), its antiderivative is \(2 \ln|t|\).
• The term \(-\frac{3}{t^2}\) is integrated by recognizing it as \(-3t^{-2}\), leading to an antiderivative \(\frac{3}{t}\).

Each of these operations results in a function describing cumulative change, and the constant \(C\) is added as a reminder that integration determines a family of functions.

Polynomial Expansion

Polynomial expansion is a specific type of algebraic manipulation where products of binomials or other polynomial terms are expanded into an extended expression. This expansion is significant in calculus as it often unveils simpler terms that match straightforward integration rules. When expanding \((t-1)(t+3)\),

• you multiply every term in the first binomial by each term in the second binomial.
• This results in: \(t \cdot t + t \cdot 3 - 1 \cdot t - 1 \cdot 3\),
• which simplifies to \(t^2 + 2t - 3\).

Polynomial expansions transform complex expressions, allowing us to use basic calculus techniques effectively. Recognizing when and how to expand polynomials is critical in handling integrals more intuitively, clearing obstacles on the path toward integration.