Step 1: Understanding the Question:
This is a standard trigonometric integration problem. The integrand contains terms in the denominator that make direct integration difficult.
The goal is to transform the integrand using trigonometric identities into basic forms that can be integrated directly.
Step 2: Key Formula or Approach:
1. Pythagorean identity: \( 1 = \sin^2 x + \cos^2 x \).
2. Standard integrals: \( \int \sec^2 x dx = \tan x + c \) and \( \int \text{cosec}^2 x dx = -\cot x + c \).
Step 3: Detailed Explanation:
Start by substituting the constant 1 in the numerator with \( \sin^2 x + \cos^2 x \):
\[ \int \frac{1}{\sin^2 x \cos^2 x} dx = \int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} dx \]
Split the integral into two separate fractions:
\[ \int \left(\frac{\sin^2 x}{\sin^2 x \cos^2 x} + \frac{\cos^2 x}{\sin^2 x \cos^2 x}\right) dx \]
Simplify each term by canceling common factors:
\[ \int \left(\frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}\right) dx \]
Use reciprocal identities:
\[ \int (\sec^2 x + \text{cosec}^2 x) dx \]
Integrate each term independently:
\[ \int \sec^2 x dx = \tan x \]
\[ \int \text{cosec}^2 x dx = -\cot x \]
Combining the results and adding the constant of integration \( c \):
\[ \tan x - \cot x + c \]
Step 4: Final Answer:
The integral is \( \tan x - \cot x + c \).