91Ó°ÊÓ

To understand the integral of the tangent function, we first need to know what the tangent function is. The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. In terms of the unit circle, it's the y-coordinate divided by the x-coordinate for a given angle.

Mathematically, the tangent function is defined as: \(\tan x = \frac{\text{sin} x}{\text{cos} x}\)
It is important to remember that the tangent function has periodic behavior with a period of \[ \frac{\text{\text{pi}}}{\text{4}} \]. Moreover, \(\tan x\) has vertical asymptotes at \(\frac{\text{\text{pi}}}{\text{2}} + k\text{\text{pi}} \) where \ k \ is any integer. This properties help us understanding graphically why the function behaves in a particular way through various intervals.

Using trigonometric identities helps us simplify and solve integrals involving trigonometric functions. Many trigonometric integrals are complex, and identifying the right identity to use makes a huge difference. Here, to solve the integral of \(\tan x\), we make use of the identity which express tangent in terms of sine and cosine:

\( \tan x = \frac{\text{sin} x}{\text{cos} x}\)
This identity transforms the integral into a more manageable format:

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