## What is The Integration of Tan x?

To understand this, let's take an example. Let's integrate the function f(x) = tan(x) with respect to x.

We will begin by using the substitution method, where we make the substitution u = cos(x) and du = -sin(x)dx. This substitution is made because the derivative of cos(x) is -sin(x) and the function tan(x) = sin(x)/cos(x).

So, the integral becomes:

∫f(x)dx = ∫tan(x)dx = ∫(sin(x)/cos(x))dx = ∫(-du/u) = ln|u| + C = ln|cos(x)| + C = ln|sec(x)| + C

Here C is constant of integration and sec(x) = 1/cos(x)

So the definite integration of tan(x) from x = a to x = b would be ln|sec(b)| - ln|sec(a)| + C

For example, if we want to find the integral of tan(x) from x = 0 to x = pi/4, we would have:

∫tan(x)dx from x = 0 to x = pi/4 = ln|sec(pi/4)| - ln|sec(0)| + C = ln(√2) - (-∞) + C

And the value of the constant of integration C can be found by evaluating definite integral with definite limits and the known value of the function.