Problem 37 - Calculus

Question

Evaluate \(37-42\) by any method. $$ \int_{5}^{10} \frac{d t}{t}-\int_{5 x}^{10 x} \frac{d t}{t} $$

Step-by-Step Solution

Answer

The value is 0.

Step 1: Simplify the Integrals

First, evaluate the given definite integrals. Notice that the integrand \( \frac{1}{t} \) is the integrand of the natural logarithm. Thus, the integral \( \int \frac{dt}{t} \) is equal to \( \ln|t| + C \), where \( C \) is a constant.

Step 2: Evaluate the First Integral

Compute \( \int_{5}^{10} \frac{dt}{t} \). Using the fundamental theorem of calculus, we find that:\[\int_{5}^{10} \frac{dt}{t} = \ln|10| - \ln|5| = \ln\left(\frac{10}{5}\right) = \ln(2).\]

Step 3: Evaluate the Second Integral

Now, compute \( \int_{5x}^{10x} \frac{dt}{t} \). The evaluation gives:\[\int_{5x}^{10x} \frac{dt}{t} = \ln|10x| - \ln|5x| = \ln\left(\frac{10x}{5x}\right) = \ln(2).\]

Step 4: Subtract the Results

Subtract the result of the second integral from the first:\[\ln(2) - \ln(2) = 0.\]

Step 5: Final Evaluation

The subtraction of the two integrals yields zero as both integrals evaluate to \(\ln(2)\). Therefore, the final answer is \(0\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). This is a special constant known as Euler's number. The natural logarithm is the inverse operation of exponentiation with base \( e \). When you hear "natural logarithm," think about the operation needed to undo exponentiating a number with \( e \).
This concept is particularly important in the context of calculus and integration. For instance, the integral \( \int \frac{1}{t} \, dt \) is equal to \( \ln|t| + C \), where \( C \) is the constant of integration. This property arises because taking the derivative of \( \ln|t| \) gives \( \frac{1}{t} \), aligning with the rules of differentiation. Understanding this relationship is crucial for solving problems involving integrals with this specific integrand.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone in the study of calculus, linking the concept of differentiation with integration. It has two main parts:

• Part 1: If a function \( f \) is continuous over an interval \([a, b]\), and \( F \) is an antiderivative of \( f \) on \([a, b]\), then the integral of \( f \) from \( a \) to \( b \) is given by \( F(b) - F(a) \).
• Part 2: If \( F \) is defined by \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \).

The exercise utilizes the first part of this theorem. We calculate definite integrals such as \( \int_{5}^{10} \frac{dt}{t} \) using the formula \( F(b) - F(a) \), ensuring \( F(t) \) is a known antiderivative, namely \( \ln|t| \). Each definite integral is computed over its specified interval, using the boundary values to find the "net area" under the curve.

Integrand

The term "integrand" refers to the function inside the integral, the part we aim to integrate. In the problem, our integrand is \( \frac{1}{t} \). It dictates how the integral should be computed. The choice of the integrand influences the complexity and method of solving the integral.
Integrands can take many forms, but when they are simple like \( \frac{1}{t} \), known properties apply directly. Here, understanding that \( \frac{1}{t} \) integrates to a natural logarithm, \( \ln|t| \), streamlines the solution significantly. This knowledge allows us to apply integration techniques more efficiently, focusing on calculating the integral's limits. In essence, the integrand is central to determining the structure and solution method of any integral-based problem.