Question:
\[ \int \frac{\mathrm{d} x}{\sin ^2 x \cos ^2 x}= \]
\[ \int \frac{\mathrm{d} x}{\sin ^2 x \cos ^2 x}= \]
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Write the numerator 1 as sin squared plus cos squared and split the fraction.
Updated On: Nov 13, 2026
\(\mathrm{tan}x+\mathrm{cot}x+\mathrm{c}\), where c is the constant of integration.
\(\mathrm{tan}x-\mathrm{cot}x+\mathrm{c}\), where c is the constant of integration.
\(\mathrm{tan}x\mathrm{cot}x+\mathrm{c}\), where c is the constant of integration.
\(\mathrm{tan}x-\mathrm{cot}2x+\mathrm{c}\), , where c is the constant of integration.
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
Use the identity \(\sin^{2}x+\cos^{2}x=1\) in the numerator.
Step 2: Split the integrand
\(\dfrac{1}{\sin^{2}x\cos^{2}x}=\dfrac{\sin^{2}x+\cos^{2}x}{\sin^{2}x\cos^{2}x}=\sec^{2}x+\operatorname{cosec}^{2}x\)
Step 3: Calculation
\(\int\sec^{2}x\,dx+\int\operatorname{cosec}^{2}x\,dx=\tan x-\cot x+c\)
Final Answer:
The integral is \(\tan x-\cot x+c\). \[ \boxed{\text{(B) }\tan x-\cot x+c} \]
Use the identity \(\sin^{2}x+\cos^{2}x=1\) in the numerator.
Step 2: Split the integrand
\(\dfrac{1}{\sin^{2}x\cos^{2}x}=\dfrac{\sin^{2}x+\cos^{2}x}{\sin^{2}x\cos^{2}x}=\sec^{2}x+\operatorname{cosec}^{2}x\)
Step 3: Calculation
\(\int\sec^{2}x\,dx+\int\operatorname{cosec}^{2}x\,dx=\tan x-\cot x+c\)
Final Answer:
The integral is \(\tan x-\cot x+c\). \[ \boxed{\text{(B) }\tan x-\cot x+c} \]
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