Problem 110 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(107-110.\) $$\frac{1}{\ln a} \int_{1}^{x} \frac{1}{t} d t, \quad x>0$$

Step-by-Step Solution

Answer

\(\frac{\ln(x)}{\ln(a)}\)

Step 1: Recognize the Integrand

Identify the function to be integrated: \( \frac{1}{t} \). This function is known as the natural logarithm function's derivative, \( \frac{d}{dt} [\ln(t)] = \frac{1}{t} \).

Step 2: Set Up the Integral

Set up the integral as given in the exercise: \( \int_{1}^{x} \frac{1}{t} dt \) with the constant prefactor \( \frac{1}{\ln a} \).

Step 3: Apply the Fundamental Theorem of Calculus

Use the Fundamental Theorem of Calculus to evaluate the integral \( \int_{1}^{x} \frac{1}{t} dt = \ln(x) - \ln(1) \). Note that \( \ln(1) = 0 \).

Step 4: Simplify the Result

Substitute the evaluated integral back into the expression: \( \frac{1}{\ln a} (\ln(x) - 0) = \frac{1}{\ln a} \ln(x) \).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration. It comprises two main parts:

• The first part ensures that if you have an antiderivative of a function, you can use it to evaluate definite integrals.
• The second part explains how to construct an antiderivative of a given function by integrating it.

In this context, we use it to evaluate the integral of function \( \frac{1}{t} \).
We apply it to obtain \( \ln(x) - \ln(1) \) since the derivative of \( \ln(t) \) is \( \frac{1}{t} \).
The theorem simplifies calculating areas under curves and shows how integration can essentially "undo" differentiation.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with the base \( e \) (approximately 2.718).
It's the inverse of the exponential function. The natural log answers the question: "To what power must \( e \) be raised to obtain \( x \)?"
This function has an important derivative: \( \frac{d}{dt} [\ln(t)] = \frac{1}{t} \).

• The derivative tells us the rate of change of \( \ln(t) \) with respect to \( t \).
• Antiderivatives or integrals involving \( \frac{1}{t} \) naturally involve the \( \ln(t) \) function.

The derivative relationship is crucial in solving integrals like \( \int \frac{1}{t} dt \), leading directly to \( \ln(t) + C \), where \( C \) is the integration constant.

Integration Techniques

Integration techniques refer to the many methods used to find the antiderivative or the integral of functions.
For the integral \( \int_{1}^{x} \frac{1}{t} dt \), recognizing the integrand as the derivative of a known function is key.

• Understanding that \( \frac{1}{t} \) is the derivative of \( \ln(t) \) simplifies the problem significantly.
• Applying the Fundamental Theorem of Calculus helps easily evaluate the integral over a specified interval.

Once evaluated, inserting it into the original expression is a simple algebraic manipulation.
This technique underlines the importance of recognizing common integrals and their corresponding antiderivatives.