Evaluate the following integrals:

Let

We know (a + b)² = a² + 2ab + b²

Therefore, we can write the integral as

We have sec²θ – tan²θ = cosec²θ – cot²θ = 1

Recall and

We also have

⇒ I = a²tan x + b²(–cot x) – (a - b)² × x + c

∴ I = a²tan x – b²cot x – (a – b)²x + c

Thus,