Question

For the following exercises, find the indefinite integral. $$ \int \frac{d t}{3 t} $$

Short answer

\( \frac{1}{3} \ln |t| + C \)

Step-by-step solution

Identify the Integral Type

The given integral is in the form \( \int \frac{1}{t} dt \), which suggests that it can directly use the integral rule for \( \frac{1}{t} \). To simplify, notice that \( \frac{1}{3t} = \frac{1}{3} \cdot \frac{1}{t} \).

Factor Out Constant Multipliers

According to integral rules, constants can be factored out of the integral. Thus, \( \int \frac{1}{3t} dt = \frac{1}{3} \int \frac{1}{t} dt \).

Integrate using the \\ln Function

The integral \( \int \frac{1}{t} dt \) is a known form, and its integral is \( \ln |t| + C \), where \( C \) is the constant of integration. Thus, \( \frac{1}{3} \cdot \int \frac{1}{t} dt = \frac{1}{3} \ln |t| + C \).

Combine the Results

Attaching the constant \( \frac{1}{3} \) to the result from the integral, we obtain the indefinite integral as \( \frac{1}{3} \ln |t| + C \).

Key concepts

Integration Techniques

Integration is the process opposite of differentiation and is used to find functions when their derivatives are known. There are various techniques that can be used, each suitable for different types of integrals:

• **Basic Integration**: Useful when dealing with straightforward functions like powers of x, trigonometric functions, and others.
• **Integration by Substitution**: Applied when the integral resembles the derivative of a composite function.
• **Integration by Parts**: Based on the product rule for differentiation and useful for products of functions.
• **Partial Fraction Decomposition**: Used for rational functions to break them into simpler fractions.

In the exercise provided, the integration technique used is recognizing the integral of form \( \int \frac{1}{t} dt \). This falls under basic integration with the result being the natural logarithm function, \( \ln |t| \). Remember, constants like \( \frac{1}{3} \) can be factored out of the integral, simplifying the operation: \[ \int \frac{1}{3t} dt = \frac{1}{3} \int \frac{1}{t} dt. \]This step simplifies your work and highlights the importance of recognizing patterns in integrals.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a useful mathematical function that is the inverse of the exponential function with base \( e \), where \( e \approx 2.71828 \). When we integrate \( \frac{1}{t} \), the result is the natural logarithm of the absolute value of \( t \): \[ \int \frac{1}{t} dt = \ln |t| + C. \]A few important properties of natural logarithms include:

• \( \ln(ab) = \ln(a) + \ln(b) \)
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• \( \ln(a^b) = b \cdot \ln(a) \)

In problem-solving, natural logarithms arise frequently, especially in integration problems involving \( \frac{1}{x} \). They are essential for simplifying expressions and solving equations.

Constant of Integration

When dealing with indefinite integrals, one crucial component is the **constant of integration,** usually denoted by \( C \). This constant represents the family of all antiderivatives, indicating that there are infinitely many functions that could satisfy the integral. Here's why the constant is needed:

• Integrals reverse differentiation, and when differentiating a constant, the result is zero. Thus, an indefinite integral can only be determined up to a constant.
• In any real-world interpretation, this constant can represent initial conditions or a starting point for calculations, giving full meaning to the antiderivative.

For the indefinite integral \( \int \frac{1}{3t} dt \), the result is \( \frac{1}{3} \ln |t| + C \), where \( C \) is added to incorporate all possible solutions.