The integral of tan(x) is given by ln|sec(x)| + C or -ln|cos(x)| + C. Here, C is the constant of integration. The function tan(x) is continuous at all real numbers, except x = (2n+1)π/2, where n is an integer. To find the integral of tan(x), we can express it in terms of sine and cosine so that it becomes an integrable function. As per the definition of tan(x), we have tan(x) = sin(x) / cos(x). This can be rewritten as ∫ tan(x) dx = ∫ (sin(x) / cos(x)) dx = ∫ (1/cos(x)) sin(x) dx. We can then apply the substitution method of integration by letting u = cos(x), which gives us du = -sin(x) dx. Substituting these values, we get ∫ tan(x) dx = ∫ (sin(x) / cos(x)) dx = ∫ (1/u) (-du) = -∫ (1/u) du = -ln|u| + C = -ln|cos(x)| + C = ln|sec(x)| + C.