What is the integral of tan(x) dx?

The integral of tan(x) with respect to x is indeed represented as -ln|cos(x)| + C. To understand why this is the correct result, it's important to consider the relationship between the integral of tan(x) and its trigonometric identity.

Recall that tan(x) can be expressed as sin(x)/cos(x). Thus, we can rewrite the integral:

∫ tan(x) dx = ∫ (sin(x)/cos(x)) dx.

To solve this integral, we can use a substitution method. Let u = cos(x). Then, the differential du is given by du = -sin(x) dx, which means that -du = sin(x) dx.

Substituting these into the integral, we have:

∫ (sin(x)/cos(x)) dx = ∫ (-1/u) du,

which simplifies to:

• ln|u| + C = -ln|cos(x)| + C.

This shows that -ln|cos(x)| + C is the correct expression for the integral of tan(x).

While the other choices may seem plausible at first glance, they do not correctly derive from the process of integration applied to tan(x). The choice involving sec(x) or sin(x) does