What is the integral of tan(x) dx?

To find the integral of (\tan(x) dx), we can use the relationship between tangent and sine and cosine. Recall that (\tan(x) = \frac{\sin(x)}{\cos(x)}).

To integrate (\tan(x)), we can rewrite it as:

[

\int \tan(x) dx = \int \frac{\sin(x)}{\cos(x)} dx.

]

A common method to integrate this function is to use substitution. Let (u = \cos(x)). Then, the derivative of (u) with respect to (x) is (-\sin(x) dx), which implies that (dx = -\frac{du}{\sin(x)}).

Substituting these into the integral, we can write:

[

\int \tan(x) dx = -\int \frac{1}{u} du.

]

The integral of (\frac{1}{u}) is (\ln|u| + C), leading us to:

[

-\ln|u| + C = -\ln|\cos(x)| + C.

]

Thus, the integral of (\tan(x) dx)