Question

Prove that the function \(F(x)=\int_{x}^{2 x} \frac{1}{t} d t\) is constant on the interval \((0, \infty)\).

Short answer

The function \(F(x)=\int_{x}^{2x} \frac{1}{t} dt\) is not constant on the interval \((0, \infty)\) as its derivative \(F'(x) = \ln(2)\) does not equal to zero over this interval.

Step-by-step solution

Derive the function

By using the Fundamental Theorem of Calculus, derive the function \(F(x)\). The derivative of an integral of a function from one limit that is a function of \(x\) to another limit that is a function of \(x\) can be found by subtracting the antiderivative of the limit evaluated at the upper bound and the antiderivative of the limit evaluated at the lower bound. For the integral \(\int_{x}^{2x} \frac{1}{t} dt\), the antiderivative of \(\frac{1}{t}\) is \(\ln |t|\). Thus, the formula gives \(F'(x) = \ln |2x| - \ln |x|\).

Simplify the expression

Simplify the difference of the logarithms according to the rules of logarithms \(\ln a - \ln b = \ln \left( \frac{a}{b}\right)\). Thus the expression simplifies to \(F'(x) =\ln \left(\frac{2x}{x}\right)\). Further simplifying the expression inside the logarithm yields \(F'(x) = \ln(2)\).

Evaluate the derivative

Evaluate the derivative \(F'(x)\). Since \(\ln(2)\) is a constant, we have that \(F'(x) = \ln(2)\), for every \(x\) in the open interval \((0, \infty)\). This means the derivative does not equal to zero, so the original function is not constant on the interval.

Key concepts

Integration

Integration is a fundamental concept in calculus that refers to the process of finding the integral of a function. It is the reverse operation of differentiation, where we determine the area under a curve or, more generally, finding the total accumulation of quantities.

When we speak of \textbf{definite integration}, which is what happens in the exercise above, we're referring to the accumulation of a function's values between two specific points on the x-axis—these are our limits of integration. In the exercise, the integral \(\int_{x}^{2x} \frac{1}{t} dt\) calculates the area under the curve of the function \(f(t) = \frac{1}{t}\) from \(x\) to \(2x\), where \(x\) is within the interval \( (0, \(\infty\)) \).

Understanding the properties of integrals is essential for various applications in physics, engineering, and economics, where calculating total values, such as distance, area, and other quantities is crucial.

Derivative of an Integral

The derivative of an integral may sound like a daunting task, but it is an elegant concept explained by the \textbf{Fundamental Theorem of Calculus}. This theorem bridges the two central operations in calculus: differentiation and integration. It tells us that if you start with a continuous function, integrate it over an interval, and then take the derivative of this integral, you are essentially back to your original function.

In our textbook example, when you derive the function \(F(x)\), the theorem helps us to do so by decomposing the integral into differences of its antiderivative evaluated at the upper and lower limits, as shown in the solution steps. A significant insight from the Fundamental Theorem of Calculus is that the integral's derivative can reveal how changes at the boundaries of integration impact the accumulated value. This correlation is profoundly useful in both theoretical and practical areas of mathematics and science.

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions, and they play a critical role in the world of mathematics because of their unique properties. The logarithm function, usually written as \(\log_b(x)\) or for the natural logarithm \(\ln(x)\), tells us the power to which the base, \(b\), must be raised to obtain \(x\).

In the context of our problem, when you deal with the \textbf{natural logarithm \(\ln\)}, it means that the base \(b\) is equal to \(e\), which is approximately 2.71828. This constant \(e\) is a fundamental mathematical constant and is particularly important in calculus.

In the solution process, understanding the properties of logarithms is key to simplifying expressions, such as \(\ln a - \ln b = \ln(\frac{a}{b})\). In the exercise, this logarithmic identity was crucial to simplify the difference of two logarithms and to reveal that \(F'(x) = \ln(2)\), indicating a constant slope of our original function over the interval in question.