Question

$\int \frac{dx}{\sin^{2}x \cos^{2}x} =$

• A. $\tan x + \cot x + c$
• B. $\tan x - \cot x + c$
• C. $\tan x \cot x + c$
• D. $\tan x + \sec x + c$

Hint:

When you see $\sin^2 x \cos^2 x$ in the denominator, adding "1" (as $\sin^2 + \cos^2$) in the numerator is the standard trick.

Answer 1

The Correct Option is B

Step 1: Concept
Use the trigonometric identity $1 = \sin^2 x + \cos^2 x$ in the numerator.

Step 2: Meaning
Splitting the integral makes it easier to integrate using standard forms.

Step 3: Analysis
$\int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} dx = \int (\frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}) dx = \int (\sec^2 x + \csc^2 x) dx$.

Step 4: Conclusion
The integrals are $\tan x$ and $-\cot x$ respectively. Thus, $\tan x - \cot x + c$.
Final Answer: (B)