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The Fundamental Theorem of Calculus is a pivotal concept in calculus that bridges the gap between differentiation and integration. It comes in two main parts: the first connects the operation of differentiation to that of integration, and the second is particularly useful in finding the value of a definite integral when an antiderivative is known.
For a definite integral of the form \( \int_{a}^{b} f(x) \, dx \), the theorem states that the integral can be evaluated by using an antiderivative \( F \) of \( f \).
Mathematically, this is represented as:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Where \( F(b) \) and \( F(a) \) are the evaluations of the antiderivative \( F \) at the upper limit \( b \) and lower limit \( a \), respectively.

• This theorem simplifies the process of calculating the area under a curve, making it much more efficient.
• It highlights the integral's dependency on the upper and lower limits and the function's behavior in the interval.
• The Fundamental Theorem is crucial for connecting different parts of calculus into a unified theory.

An antiderivative, in calculus, is a function that reverses the process of differentiation. Finding an antiderivative of a given function is essentially the opposite of finding its derivative.
If the derivative of a function \( F(x) \) is \( f(x) \), then \( F(x) \) is called an antiderivative of \( f(x) \).
For the integral in our exercise, the function \( f(x) = \frac{\ln x}{x} \) requires finding an antiderivative. We note that one antiderivative of this function is given by:
\[ F(x) = \frac{1}{2} (\ln x)^2 \]
This means that the derivative of \( \frac{1}{2} (\ln x)^2 \) will give us back \( \frac{\ln x}{x} \), confirming its role as an antiderivative.

• Antiderivatives can help find integrals, especially in the context of the Fundamental Theorem of Calculus.
• Understanding how to determine antiderivatives is essential for solving indefinite integrals.
• Different functions might have multiple antiderivatives, but they differ by a constant.

The natural logarithm is an essential mathematical function often denoted by \( \ln x \). It is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828.
Natural logarithms are particularly important in calculus, especially due to their simple derivative:
\[ \frac{d}{dx} (\ln x) = \frac{1}{x} \]
In our exercise, the integral involves the function \( \frac{\ln x}{x} \), capitalizing on the relationship between \( \ln x \) and its derivative. This relationship is key in the process of integration, particularly when using substitution or finding antiderivatives.

• Natural logarithms are commonly used in calculus due to their unique properties and natural derivatives.
• They frequently appear in problems requiring integration and differentiation.
• The properties of logarithms, like the power rule \( \ln(x^a) = a \ln x \), play a crucial role in simplifying calculations.

The substitution method is a technique used in integration to simplify integrals by changing the variable of integration. It is often employed when the integrand is a composite function, making it difficult to solve directly.
This method involves the following steps, often likened to the chain rule in reverse:

• Select a substitution, \( u = g(x) \), where \( g(x) \) is a part of the integrand.
• Calculate \( \frac{du}{dx} \) or equivalently, \( du = g'(x) \, dx \), to make the integral easier.
• Rewrite the integral in terms of \( u \) and \( du \).
• Solve the integral in terms of \( u \), and finally, substitute back to express the solution in terms of \( x \).

In the example given, recognizing that \( u = \ln x \) simplifies the integration process due to the inherent derivative relationship \( du = \frac{1}{x} \, dx \), making it easier to evaluate the definite integral. Using substitution transforms complex integrals into basic ones that are more manageable.