How to use integration rules to evaluate the integral of tan (2x)?

First, we'll substitute 2x by
t.

2x = t

Differentiating both
sides, we'll get:

2dx = dt => dx =
dt/2

We'll re-write the
integral:

Int tan(2x)dx = Int (tan
t)*dt/2

We'll write tan t = sin t/cos
t

Int (tan t)*dt/2 = Int (sin t/cos
t)*(dt/2)

We'll apply substitution technique to solve the
indefinite integral. We'll put cos t = v.

- sin t dt =
dv

Int (sin t/cos t)*(dt/2) = Int -dv/2v = - (1/2)*ln |v| +
C

Int (tan t)*dt/2 = - (1/2)*ln |cos t| +
C

Int tan (2x)*dx = - (1/2)*ln |cos (2x)| +
C

Int tan (2x)*dx = ln [1/sqrt (cos 2x)] +
C

The indefinite integral of the given
function is: Int tan (2x)*dx = ln [1/sqrt (cos 2x)] +
C