Question

Evaluate the integral: \[ \int \frac{1}{\sin^2 2x \cdot \cos^2 2x} \, dx \]

• A. \( \frac{1}{2} \tan 2x \)
• B. \( \frac{1}{2} \cot 2x \)
• C. \( \frac{1}{4} \cot 2x \)
• D. \( \frac{1}{4} \tan 2x \)

Hint:

When encountering integrals involving \( \cot^2 x \), rewrite them in terms of \( \csc^2 x \) to make them easier to solve.

Answer 1

The Correct Option is B

We are tasked with evaluating the integral: \[ \int \frac{1}{\sin^2 2x \cdot \cos^2 2x} \, dx \]
Step 1: Use trigonometric identity We can start by simplifying the integrand using the identity: \[ \sin^2 A \cdot \cos^2 A = \frac{1}{4} \sin^2 2A \] Using this identity, we rewrite the integral: \[ \int \frac{1}{\sin^2 2x \cdot \cos^2 2x} \, dx = \int \frac{4}{\sin^2 4x} \, dx \]
Step 2: Express in terms of \( \cot \) We now recognize that \( \frac{1}{\sin^2 A} \) can be rewritten as \( \cot^2 A \), so the integral becomes: \[ \int 4 \cdot \cot^2 4x \, dx \]
Step 3: Use standard integral formula We use the standard integral formula for \( \cot^2 x \): \[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx \] This simplifies to: \[ \int 4 \cdot (\csc^2 4x - 1) \, dx \]
Step 4: Solve the integral Breaking it into two integrals, we have: \[ 4 \int \csc^2 4x \, dx - 4 \int 1 \, dx \] The integral of \( \csc^2 x \) is \( -\cot x \), so: \[ 4 \left( -\frac{1}{4} \cot 4x \right) - 4x = -\cot 4x - 4x \] Thus, the solution to the integral is: \[ \boxed{-\cot 4x - 4x} \]