Question:
The value of \( \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x}\,dx \) is equal to
The value of \( \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x}\,dx \) is equal to
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Look for symmetry or complementary integrals to simplify definite integrals.
Updated On: Apr 21, 2026
- \( \frac{\pi}{20} \)
- \( \frac{\pi}{10} \)
- \( \frac{3\pi}{20} \)
- \( \frac{\pi}{5} \)
- \( \frac{\pi}{4} \)
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The Correct Option is C
Solution and Explanation
Concept:
Use identity:
\[
\frac{\cot^3 x}{1+\cot^3 x} + \frac{1}{1+\cot^3 x} = 1
\]
Step 1: Split integral.
\[ I + J = \int_{\pi/10}^{2\pi/5} dx = \frac{3\pi}{10} \]
Step 2: Symmetry.
\[ I = J \Rightarrow 2I = \frac{3\pi}{10} \Rightarrow I = \frac{3\pi}{20} \]
Step 1: Split integral.
\[ I + J = \int_{\pi/10}^{2\pi/5} dx = \frac{3\pi}{10} \]
Step 2: Symmetry.
\[ I = J \Rightarrow 2I = \frac{3\pi}{10} \Rightarrow I = \frac{3\pi}{20} \]
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