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The tangent function, often denoted as \( \tan(x) \), represents a crucial concept in trigonometry. It is the ratio of the sine to the cosine of an angle. In simpler terms, for a given angle \( x \), the tangent function is defined as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

The tangent function has some intriguing properties:

• Its graph shows repeating patterns called periods, with a typical period of \( \pi \) radians.
• As an odd function, it is symmetrical about the origin; thus, \( \tan(-x) = -\tan(x) \).
• It has vertical asymptotes wherever the cosine of \( x \) is zero, due to division by zero.

Understanding these properties helps in calculating integrals involving the tangent function, as being familiar with its pattern and limitations guides us effectively through the integration process. By converting the function into another that is easier to handle, such as via substitution, you can integrate functions including the tangent.

Integration by substitution is a powerful technique to solve complex integrals. Essentially, it simplifies integrals by transforming them into a more workable form. This involves substituting part of the integral with a new variable, which often turns the integral into something more straightforward.
In this exercise, the substitution involved setting \( u = 2x \) and hence \( du = 2dx \), which transforms the given integral into:

• \( \int \tan 2x \, dx = \frac{1}{2} \int \tan u \, du \)

By using substitution, we replace the variable of integration from \( x \) to \( u \), making it easier to apply known integration formulas.
This method shines particularly in cases where the integrand (the function being integrated) transforms into a familiar form, allowing us to apply basic integration formulas and then revert using the original variables.

A definite integral computes the accumulation of quantities, like areas under curves, between specific limits. The integral from \( a \) to \( b \) of a function \( f(x) \) is denoted by the symbol:

• \( \int_{a}^{b} f(x) \, dx \)

In this exercise, the definite integral to evaluate was \( \int_{0}^{\pi/8} \tan 2x \, dx \). After finding the antiderivative, the definite integral is calculated by:

• Evaluating the antiderivative at the upper limit (\( x = \pi/8 \)).
• Evaluating the antiderivative at the lower limit (\( x = 0 \)).
• Subtracting the lower limit result from the upper limit result.

This results in a single numerical value, providing a finite measure of the accumulated area (or other quantitative accumulations) between the limits \( a \) and \( b \). It is an integral part of integral calculus, widely used in physics, engineering, and various applied sciences.

Indefinite integrals are solutions to differential equations and represent a family of functions. Unlike definite integrals, they don't have set limits and include an added constant \( C \), referred to as the constant of integration.
For the tangent function, integrating \( \int \tan u \, du \) provided us the result as:

• \( \ln |\sec u| + C \)

This result includes \( C \) because there are infinitely many antiderivatives differing only by a constant. The indefinite integral captures this idea:

• Representing general solutions rather than specific values.
• Including the constant acknowledges any number of vertical shifts of the elementary antiderivative function.

In problems like these, indefinite integrals are used to find the general form of solutions, subsequently refined into definite integrals when specific limits and conditions are provided.