How do you integrate tan x?

Solution

We know that:

tan x = sin x / cos x

So,

âˆ« tan x dx = âˆ« (sin x / cos x) dx

Step 1: Substitution

Let:

u = cos x

Then:

du/dx = -sin x

du = -sin x dx

Step 2: Substitute in the Integral

âˆ« (sin x / cos x) dx = - âˆ« (1 / u) du

Step 3: Integrate

- âˆ« (1 / u) du = - ln |u| + C

Substitute back u = cos x:

- ln |cos x| + C

Step 4: Write in Standard Form

- ln |cos x| = ln |sec x|

Final Answer

âˆ« tan x dx = ln |sec x| + C

where C is the constant of integration.

Solution