Step 1: Understanding the Question:
This is an integration problem where the result is given in terms of unknown coefficients \( A \) and \( B \).
By performing the integration ourselves, we can compare our result with the provided form to find the values of the constants.
Step 2: Key Formula or Approach:
1. Split the fraction into two parts.
2. Standard integral rules for trigonometric functions.
Step 3: Detailed Explanation:
Split the integrand:
\[ \int \left(\frac{\sin^3 x}{\sin^2 x \cos^2 x} + \frac{\cos^3 x}{\sin^2 x \cos^2 x}\right) dx \]
Simplify each term by canceling common factors:
\[ \int \left(\frac{\sin x}{\cos^2 x} + \frac{\cos x}{\sin^2 x}\right) dx \]
Rewrite as products of reciprocal functions:
\[ \int \left(\frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} + \frac{\cos x}{\sin x} \cdot \frac{1}{\sin x}\right) dx \]
\[ \int (\tan x \sec x + \cot x \text{cosec } x) dx \]
Use standard integration formulas:
\( \int \sec x \tan x dx = \sec x \)
\( \int \text{cosec } x \cot x dx = -\text{cosec } x \)
The total integral result is:
\[ \sec x - \text{cosec } x + c \]
Compare this with the given form \( A \sec x + B \text{cosec } x + c \):
Comparing coefficients: \( A = 1 \) and \( B = -1 \).
The pair \( (A, B) \) is \( (1, -1) \).
Step 4: Final Answer:
The values of the constants are \( A = 1 \) and \( B = -1 \).