Find the antiderivative of tan² (x) dx?

The integral of a function is an antiderivative of that function.

Answer:

We can write tan² (x) in terms of sin² x / cos² x. Let's see how.

Explanation: 

∫ tan² (x) dx = sin² x / cos² x dx

= ∫ (1 - cos² x / cos² x) dx, ∵ sin² x = 1 - cos² x

On splitting the terms, we get

∫ (1 / cos² x) - (cos² x / cos² x) dx

= ∫ sec² x dx - ∫ 1 dx, ∵ sec² x = 1 / cos² x

= tan x - x + c

Thus, the antiderivative of tan² (x) is tan x - x + c