Problem 47 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{4} \frac{d t}{1+t} $$

Step-by-Step Solution

Answer

The value is \( \ln 5 \).

Step 1: Recognize the Integral Type

The integral \( \int \frac{dt}{1+t} \) is a logarithmic integral. This is due to the form \( \frac{1}{x} \), which suggests that the antiderivative is a natural logarithm.

Step 2: Find the Antiderivative

Recall that the antiderivative of \( \frac{1}{1+t} \) is the natural logarithm, \( \ln|1+t| \). So, the indefinite integral \( \int \frac{1}{1+t} \, dt = \ln|1+t| + C \).

Step 3: Evaluate the Definite Integral

Calculate \( \int_{0}^{4} \frac{dt}{1+t} \) by using the boundaries in the antiderivative: \[ \int_{0}^{4} \frac{dt}{1+t} = \left[ \ln|1+t| \right]_{0}^{4} \].

Step 4: Substitute the Upper Limit

Substitute \( t = 4 \) into the antiderivative results: \[ \ln|1+4| = \ln 5 \].

Step 5: Substitute the Lower Limit

Substitute \( t = 0 \) into the antiderivative results: \[ \ln|1+0| = \ln 1 \].

Step 6: Subtract the Results

Subtract the result of the lower limit from the upper limit: \[ \ln 5 - \ln 1 = \ln 5 - 0 = \ln 5 \].

Key Concepts

Logarithmic Integrals

Logarithmic integrals often appear in calculus when dealing with functions that contain a variable in the denominator. These integrals take the form \( \int \frac{dt}{1+t} \), where the focus is on expressions like \( \frac{1}{x} \).
This form is special because its antiderivative leads to a natural logarithm. Understanding the nature of logarithmic integrals is crucial when faced with definite or indefinite integrals of this type.
Essentially, whenever you see an integral with \( \frac{1}{1+t} \), you can assume it's leading towards the natural logarithm, simplifying your calculations.
Such integrals are essential in various fields like engineering and physics, where they help model exponential growth and decay.

Definite Integrals

Definite integrals are integrals evaluated over a specific interval. They're essential for calculating actual values, for example, finding areas under a curve between two points.
In our exercise, the integral from 0 to 4 of \( \frac{dt}{1+t} \) is a classic example of a definite integral.
Here’s how it works:

• You find the antiderivative of the function.
• Next, plug in the upper limit (here, 4) and then the lower limit (here, 0) into the antiderivative.
• Finally, subtract the result at the lower limit from the result at the upper limit.

This gives you the area or total change derived from the integration process within those limits, making definite integrals powerful tools for real-world applications.

Antiderivatives

Antiderivatives are the reverse of derivatives. They help us find the original function from its derivative. For the integral \( \int \frac{dt}{1+t} \), the antiderivative is \( \ln|1+t| \).
To determine an antiderivative, look at the function's form and recall basic integral rules. For instance, the antiderivative of \( \frac{1}{x} \) leads to a natural logarithm because differentiation of \( \ln(x) \) yields \( \frac{1}{x} \).
Finding antiderivatives is like piecing together a puzzle of the original function.

• Practice recognizing different forms to quickly determine antiderivatives.
• Once you find the antiderivative, use it for further calculations like definite integrals.

Mastery of antiderivatives simplifies many complex calculus problems by reversing the differentiation process.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with the base \( e \), where \( e \approx 2.71828 \). It's fundamental in calculus as it frequently appears in integration and differentiation of functions.
In the context of logarithmic integrals, the natural logarithm indicates the result of integrating specific forms like \( \frac{1}{x} \). For instance, the integral \( \int \frac{1}{1+t} \, dt \) results in \( \ln|1+t| \).
The natural logarithm has special properties:

• \( \ln(1) = 0 \)
• \( \ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right) \)

These properties can simplify calculations. Remember, natural logarithms naturally arise in exponential growth models, like population dynamics and finance, highlighting their importance.