Problem 119 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{1}^{e} \frac{1}{x} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is 1.

Step 1: Identify the Integral Expression

The given integral is \( \int_{1}^{e} \frac{1}{x} \, dx \). Here, the function to be integrated is \( \frac{1}{x} \) over the interval from 1 to \( e \).

Step 2: Recognize the Function to Integrate

The function \( \frac{1}{x} \) is a well-known function whose integral is the natural logarithm function. Specifically, the indefinite integral of \( \frac{1}{x} \) is \( \ln|x| + C \), where \( C \) is the constant of integration.

Step 3: Compute the Indefinite Integral

To solve the definite integral, we need the antiderivative. The indefinite integral of \( \frac{1}{x} \) is \( \ln|x| \). We will use this to evaluate the definite integral.

Step 4: Evaluate the Definite Integral

Substitute the upper and lower bounds of the integral into the antiderivative to find the definite integral. Calculate:\[\int_{1}^{e} \frac{1}{x} \, dx = \left[ \ln|x| \right]_{1}^{e} = \ln(e) - \ln(1) \]

Step 5: Simplify the Result

Calculate the values:\( \ln(e) = 1 \) because \( e \) is the base of the natural logarithm, and \( \ln(1) = 0 \) because the logarithm of 1 to any base is 0. Therefore\[\ln(e) - \ln(1) = 1 - 0 = 1.\]

Step 6: Conclusion of the Solution

The result of the definite integral \( \int_{1}^{e} \frac{1}{x} \, dx \) is 1.

Key Concepts

Natural Logarithm

The natural logarithm is a special logarithm used frequently in mathematics. Denoted as \( \ln(x) \), it is the logarithm with the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. This function is crucial because it allows us to solve many equations involving exponential functions. For instance, when dealing with the integral \( \int \frac{1}{x} \, dx \), the natural logarithm efficiently relates the rate of change back to its original scale. A few important properties of the natural logarithm are:

• \( \ln(e) = 1 \) because any number \( x \) logged to its own power is always 1.
• \( \ln(1) = 0 \) meaning the logarithm of 1 to any base is zero.
• It grows slowly which makes it particularly valuable in calculus.

Understanding the natural logarithm helps in grasping complex calculus concepts, especially those related to growth and decay, which are common in real-world applications.

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus that represent the family of all antiderivatives of a function. When we compute an indefinite integral, we are essentially reversing the process of differentiation. The notation for an indefinite integral looks like this: \( \int f(x) \, dx \), where \( f(x) \) is the function being integrated. The solution includes a constant of integration denoted by \( C \) because when we differentiate a constant, it disappears; hence every indefinite integral involves an arbitrary constant. For example, the indefinite integral of \( \frac{1}{x} \) can be written as \( \ln|x| + C \). Indefinite integrals are crucial for:

• Finding antiderivatives which help in solving differential equations.
• Understanding the accumulation of quantities over intervals.
• Providing the foundation for definite integrals and analyzing areas under curves.

Grasping indefinite integrals and their significance allows students to solve more complex problems involving accumulation of change.

Antiderivatives

Antiderivatives, also known as primitives, are functions that reverse derivatives. If you have a function \( f(x) \), its antiderivative \( F(x) \) is determined such that \( F'(x) = f(x) \). This means that when you derive \( F(x) \), you get back the function \( f(x) \). Antiderivatives are the key to solving integrals. For the function \( f(x) = \frac{1}{x} \), its antiderivative is \( F(x) = \ln|x| + C \). This relationship is fundamental in evaluating integrals. Antiderivatives are useful because:

• They help in constructing solutions to differential equations, which describe a wide range of phenomena.
• Every problem involving the calculation of areas, volumes, and many physical quantities can be reinterpreted as a problem involving antiderivatives.

In calculus, recognizing antiderivatives quickly allows students to set up and solve integrals, thereby unraveling constants and patterns intricately related to real-life applications. Understanding antiderivatives is akin to holding the key to solve even the most challenging integration problems seamlessly.