Problem 58 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t$$

Step-by-Step Solution

Answer

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

Step 1: Split the Integral

The given integral can be split into two separate integrals. We have:\[\int \left( \frac{3}{t} - \frac{2}{t^{2}} \right) dt = \int \frac{3}{t} dt - \int \frac{2}{t^{2}} dt\]

Step 2: Integrate \( \frac{3}{t} \)

The integral of \( \frac{3}{t} \) with respect to \( t \) can be computed using the formula for integration of \( \frac{1}{t} \):\[\int \frac{3}{t} dt = 3 \int \frac{1}{t} dt = 3 \ln |t|\]

Step 3: Integrate \( -\frac{2}{t^2} \)

The term \( \frac{2}{t^2} \) can be rewritten as \( 2t^{-2} \). The integral of \( t^{n} \) is \( \frac{t^{n+1}}{n+1} \) if \( n eq -1 \):\[-\int \frac{2}{t^2} dt = -2\int t^{-2} dt = -2 \cdot \frac{t^{-1}}{-1} = \frac{2}{t}\]

Step 4: Combine the Results

Combine the results of Step 2 and Step 3, adding the constant of integration, \( C \), to get the final solution:\[\int \left(\frac{3}{t} - \frac{2}{t^2}\right) dt = 3 \ln |t| + \frac{2}{t} + C\]

Key Concepts

Integration Techniques

Integration techniques are methods used to find the antiderivative or integral of a function. This involves reversing the process of differentiation. For the integral \( \int \left( \frac{3}{t} - \frac{2}{t^{2}} \right) dt \), we use a technique known as the decomposition of integrals.

Here's how it works:

• Decomposing the Integral: Decomposition lets you handle complex integrals by breaking them into simpler parts. For instance, \( \int \left( \frac{3}{t} - \frac{2}{t^{2}} \right) dt \) is split into \( \int \frac{3}{t} dt \) and \( -\int \frac{2}{t^{2}} dt \) to simplify the process.
• Simplifying the Expression: This involves changing terms into a more integrable form. For example, transforming \( \frac{2}{t^2} \) to \( 2t^{-2} \) enhances integration.

The overall aim is to make integration as straightforward as possible, thereby easing the solving process.

Logarithmic Integration

Logarithmic integration is a specific integration technique applied when dealing with functions in the form of \( \frac{1}{t} \). It helps in finding the integral by recognizing the pattern that relates to logarithmic functions.

For integration of \( \int \frac{3}{t} dt \):

• Basic Formula: The integral of \( \frac{1}{t} \) is \( \ln |t| \). Use this insight to tackle \( \int \frac{3}{t} dt \).
• Multiplying the Constant: Multiply the constant 3 by the integral result: \( 3 \int \frac{1}{t} dt = 3 \ln |t| \).

This technique highlights the integral's logarithmic nature, allowing you to apply the power of logarithms in integration scenarios.

Power Rule for Integration

The power rule for integration provides a straightforward way to find integrals of functions with exponents, excluding the case when the exponent is -1. This rule is essential in the integration of polynomial terms.

Consider \( \int t^n dt \), which yields \( \frac{t^{n+1}}{n+1} \) for \( n eq -1 \). This rule is applied to the function \( -\int \frac{2}{t^2} dt \).

• Rewriting the Expression: Express \( \frac{2}{t^2} \) as \( 2t^{-2} \), allowing the application of the power rule.
• Applying the Power Rule: Integrate \( t^{-2} \) using the rule: \( \int t^{-2} dt = \frac{t^{-1}}{-1} \).
• Combining Constants: Multiply the integral result by -2 for integration: \( -2 \cdot \frac{t^{-1}}{-1} = \frac{2}{t} \).

The power rule simplifies integration for terms with powers, thus easing computations in finding antiderivatives.