Question:
\(\int \sec ^{2}2x\cdot \text{cosec}^{2}2xdx=\ldots\)
\(\int \sec ^{2}2x\cdot \text{cosec}^{2}2xdx=\ldots\)
Show Hint
Write sec^2 and cosec^2 using sin and cos, or use sec^2 = 1 + tan^2, then integrate.
Updated On: Nov 12, 2026
- \(\frac{1}{2}\tan x+c\)
- \(\cot 2x+\tan 2x+c\)
- \(\sec 2x+c\)
- \(\frac{1}{2}(\tan 2x-\cot 2x)+c\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Rewrite the product using \(\text{cosec}^{2}2x=1+\cot^{2}2x\), so that it splits into two standard integrals.
Step 2: Rewrite
\(\sec^{2}2x\,\text{cosec}^{2}2x=\sec^{2}2x\left(1+\cot^{2}2x\right)=\sec^{2}2x+\text{cosec}^{2}2x\).
Step 3: Integrate
\(\int\sec^{2}2x\,dx=\frac{\tan 2x}{2}\) and \(\int\text{cosec}^{2}2x\,dx=-\frac{\cot 2x}{2}\).
Step 4: Combine
The integral is \(\frac{1}{2}(\tan 2x-\cot 2x)+c\).
Final Answer:
The integral is one half of (tan 2x minus cot 2x) plus c. \[ \boxed{\text{(D) }\dfrac{1}{2}(\tan 2x-\cot 2x)+c} \]
Rewrite the product using \(\text{cosec}^{2}2x=1+\cot^{2}2x\), so that it splits into two standard integrals.
Step 2: Rewrite
\(\sec^{2}2x\,\text{cosec}^{2}2x=\sec^{2}2x\left(1+\cot^{2}2x\right)=\sec^{2}2x+\text{cosec}^{2}2x\).
Step 3: Integrate
\(\int\sec^{2}2x\,dx=\frac{\tan 2x}{2}\) and \(\int\text{cosec}^{2}2x\,dx=-\frac{\cot 2x}{2}\).
Step 4: Combine
The integral is \(\frac{1}{2}(\tan 2x-\cot 2x)+c\).
Final Answer:
The integral is one half of (tan 2x minus cot 2x) plus c. \[ \boxed{\text{(D) }\dfrac{1}{2}(\tan 2x-\cot 2x)+c} \]
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