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The integral of tan(x) dx is ln|sec(x)| + C.

To evaluate the integral of tan(x) dx, we can use the substitution method. Let u = cos(x), then du/dx = -sin(x) and dx = du/-sin(x). Substituting these into the integral, we get:

∫tan(x) dx = ∫tan(x) (-sin(x)/-sin(x)) dx

= ∫(sin(x)/cos(x)) (-du/sin(x))

= -∫du/u

= -ln|u| + C

= -ln|cos(x)| + C

= ln|sec(x)| + C

Therefore, the integral of tan(x) dx is ln|sec(x)| + C. It is important to note that the natural logarithm function is only defined for positive values, hence the absolute value in the final answer.

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