Question

The value of \( \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x}\,dx \) is equal to

• A. \( \frac{\pi}{20} \)
• B. \( \frac{\pi}{10} \)
• C. \( \frac{3\pi}{20} \)
• D. \( \frac{\pi}{5} \)
• E. \( \frac{\pi}{4} \)

Hint:

Look for symmetry or complementary integrals to simplify definite integrals.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves a definite integral that can be simplified using a property of integrals of the form \(\int_a^b f(x) dx\). The specific property used is King's Rule: \(\int_a^b f(x) dx = \int_a^b f(a+b-x) dx\). This is particularly useful when \(f(x) + f(a+b-x)\) simplifies to a constant.
Step 2: Key Formula or Approach:
1. Let \(I = \int_a^b \frac{\cot^n x}{\cot^n x + 1} dx\). 2. Apply the property \(I = \int_a^b f(a+b-x) dx\). Here, \(a=\pi/10\) and \(b=2\pi/5 = 4\pi/10\). So, \(a+b = 5\pi/10 = \pi/2\). 3. We will have \(f(a+b-x) = f(\pi/2 - x)\). 4. We know that \(\cot(\pi/2 - x) = \tan(x)\). Use this to simplify the new integral. 5. Add the two expressions for I. The sum will be a simple integral that evaluates to \(b-a\). 6. Solve for I.
Step 3: Detailed Explanation:
Let the integral be I: \[ I = \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x} dx \quad \cdots (1) \] The limits are \(a = \frac{\pi}{10}\) and \(b = \frac{2\pi}{5}\). Their sum is \(a+b = \frac{\pi}{10} + \frac{4\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2}\). Using the property \(\int_a^b f(x) dx = \int_a^b f(a+b-x) dx\), we get: \[ I = \int_{\pi/10}^{2\pi/5} \frac{\cot^3(\pi/2 - x)}{1+\cot^3(\pi/2 - x)} dx \] Since \(\cot(\pi/2 - x) = \tan(x)\), this becomes: \[ I = \int_{\pi/10}^{2\pi/5} \frac{\tan^3 x}{1+\tan^3 x} dx \] Let's rewrite \(\tan x\) as \(1/\cot x\): \[ I = \int_{\pi/10}^{2\pi/5} \frac{1/\cot^3 x}{1+1/\cot^3 x} dx = \int_{\pi/10}^{2\pi/5} \frac{1/\cot^3 x}{(\cot^3 x+1)/\cot^3 x} dx \] \[ I = \int_{\pi/10}^{2\pi/5} \frac{1}{1+\cot^3 x} dx \quad \cdots (2) \] Now, add equation (1) and equation (2): \[ I + I = \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x} dx + \int_{\pi/10}^{2\pi/5} \frac{1}{1+\cot^3 x} dx \] \[ 2I = \int_{\pi/10}^{2\pi/5} \left( \frac{\cot^3 x + 1}{1+\cot^3 x} \right) dx \] \[ 2I = \int_{\pi/10}^{2\pi/5} 1 \, dx \] \[ 2I = [x]_{\pi/10}^{2\pi/5} = \frac{2\pi}{5} - \frac{\pi}{10} = \frac{4\pi}{10} - \frac{\pi}{10} = \frac{3\pi}{10} \] Now, solve for I: \[ I = \frac{3\pi}{20} \] Step 4: Final Answer:
The value of the integral is \(\frac{3\pi}{20}\).