Question

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \] 

Hint:

When dealing with integrals that involve trigonometric identities, attempt simplifying the integrand first by using fundamental identities like \( \cot x - \tan x \), and use substitution to reduce the complexity. Numerical methods may be necessary for complex integrals.

Answer 1

Step 1: Simplify the integrand using the identity for \( \cot x - \tan x \). We know that: \[ \cot x = \frac{1}{\tan x}, \quad \text{so} \quad \cot x - \tan x = \frac{1 - \tan^2 x}{\tan x}. \] Substitute this into the integral: \[ I = \int \frac{\sec^2 2x}{\left( \frac{1 - \tan^2 x}{\tan x} \right)^2} \, dx. \] This simplifies to: \[ I = \int \frac{\sec^2 2x \cdot \tan^2 x}{(1 - \tan^2 x)^2} \, dx. \] 

Step 2: Apply a substitution to simplify the integral. Let: \[ u = \tan x, \quad du = \sec^2 x \, dx. \] The integral now becomes: \[ I = \int \frac{u^2 \cdot \sec^2 2x}{(1 - u^2)^2} \, du. \] At this point, we need a further substitution or simplification method to evaluate the integral, but due to its complexity, solving this exactly involves advanced trigonometric substitutions or numerical methods, which might be outside of simple manual calculation.