How do you integrate tan x?

Solution

We know that:

tan x = sin x / cos x

So,

∫ tan x dx = ∫ (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

∫ (sin x / cos x) dx = − ∫ (1 / u) du

Step 3: Integrate

− ∫ (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

∫ tan x dx = ln |sec x| + C

where C is the constant of integration.