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 and Its Limits

The given integral is \( \int_{1}^{e} \frac{1}{x} \, dx \), which is a definite integral from 1 to \( e \). The function we are integrating is \( \frac{1}{x} \).

Step 2: Find the Antiderivative

The antiderivative of \( \frac{1}{x} \) is \( \ln|x| + C \). For this definite integral, we can simply use \( \ln x \) since \( x \) is strictly positive over the interval from 1 to \( e \).

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the definite integral from \( a \) to \( b \) of a function is the difference between its antiderivative evaluated at \( b \) and \( a \).Thus, compute \( \ln(e) - \ln(1) \).

Step 4: Evaluate the Antiderivative at the Bounds

Calculate \( \ln(e) \) and \( \ln(1) \). Since \( \ln(e) = 1 \) (because \( e \) is the base of the natural logarithm), and \( \ln(1) = 0 \), the expression \( \ln(e) - \ln(1) \) simplifies to \( 1 - 0 \).

Step 5: Finalise the Result

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

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a crucial bridge between calculus and analysis. This theorem connects differentiation with integration, showing that they are essentially inverse processes. Here's how it works:

• It states that if you have a continuous function on a closed interval \([a, b]\), then the definite integral of this function over \([a, b]\) is related to its antiderivative.
• An antiderivative of a function \(f(x)\) is another function \(F(x)\) whose derivative is \(f(x)\), meaning \(F'(x) = f(x)\).
• The theorem asserts that the definite integral from \(a\) to \(b\) of a function \(f(x)\) is \(F(b) - F(a)\), where \(F(x)\) is the antiderivative of \(f(x)\).

This makes evaluating definite integrals simpler once an antiderivative is found, as it avoids the need to perform the integration itself each time. This principle was directly applied in the exercise, where the integral of \(\frac{1}{x}\) from 1 to \(e\) was calculated using its antiderivative, the natural logarithm.

Antiderivatives

Antiderivatives are functions that reverse the process of differentiation. If you differentiate an antiderivative, you get back the original function. Here's what to know:

• An antiderivative of a function \(f(x)\) can be expressed as \(F(x) + C\), where \(C\) is a constant, since the derivative of a constant is zero.
• In many cases, finding the correct antiderivative involves identifying the form of the original function and recalling standard integral formulas.
• For example, the antiderivative of \(\frac{1}{x}\) is \(\ln|x| + C\), leveraging the properties of logarithms.

In our exercise, the function \(\frac{1}{x}\) has a straightforward antiderivative \(\ln|x|\). Because the limits of integration are 1 and \(e\) (positive values), you can safely use \(\ln x\) over this interval, simplifying calculations.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental concept in calculus closely tied to the number \(e\). This function has properties that make it very useful in calculus:

• The base of the natural logarithm is the number \(e\), which is approximately 2.71828.
• One of the key properties is \(\ln(e) = 1\), because \(e^1 = e\).
• Another useful property is \(\ln(1) = 0\), because any number raised to the power of zero is 1.

In the exercise, the natural logarithm function simplifies the integration process considerably. By knowing the antiderivative of \(\frac{1}{x}\) is \(\ln|x|\), computing the definite integral from 1 to \(e\) becomes straightforward: simply calculate \(\ln(e) - \ln(1)\), which results in the value 1. This step leverages the mathematical properties inherent in the natural logarithm and the constant \(e\).