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Integral calculus is a fundamental branch of calculus focused on the concept of an integral, which provides a way to calculate the accumulation of quantities, such as areas under curves, total distances covered, or volumes of solids. It is often seen as the counterpart to differential calculus, which studies the rates at which quantities change.

The process of finding integrals is called integration. There are several techniques to perform this, such as substitution, integration by parts, and partial fractions. In the context of finding arc length, integration plays a pivotal role in summing up infinitesimally small segments along a curve to obtain the overall length. To understand arc length, one needs to comprehend how integration accounts for continuous accumulation along a curve and how various integration techniques simplify the process of finding a solution to an integral.

The Fundamental Theorem of Calculus bridges the two central concepts of calculus: differentiation and integration, revealing how they are inverse operations. It consists of two related parts. The first part provides a certain integral to be an antiderivative of the function being integrated. Specifically, if you have a continuous function, \( f(x) \) and you are integrating it over a range from \( a \) to \( x \) in the form \( \int_{a}^{x} f(t) dt \), the derivative of this integral with respect to \( x \), is \( f(x) \) itself. This is why in the step-by-step solution provided, after differentiating the integral function, we get the original function under the square root as the derivative, \( f'(x) = \sqrt{\ln ^{2} x-1} \).

The second part, sometimes called the Evaluation Theorem, tells us that if we want to evaluate the definite integral of a continuous function from \( a \) to \( b \), we can find any antiderivative of the function and simply subtract its values at \( b \) and \( a \) to get the result. This is especially useful for calculating areas under curves or, as in our case, the arc length of a function.

Integration by parts is a technique used to integrate the product of two functions. It is derived from the product rule for differentiation and is formally stated by the equation \( \int u dv = uv - \int v du \). In practice, it involves choosing which function in the integral to set as \( u \) (which you differentiate), and \( dv \) (which you integrate), to simplify the computation.

In the exercise provided, this method is necessary to tackle the integral of \( u e^u \), since it involves the product of \( u \) and \( e^u \). To apply integration by parts, the student sets \( u \) to \( \ln x \) and differentiates it, while \( dv \) is the remaining part of the function, which in turn is integrated. The aim is to transform the original integral into a simpler form that can be easily evaluated, eventually leading to the expression \( 2e^{3} - e \) for the arc length.