Question

Evaluate the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \]

• A. \( \dfrac{\pi}{2} \)
• B. \( \dfrac{\pi}{4} \)
• C. \( \dfrac{\pi}{8} \)
• D. \( \pi \)

Hint:

For integrals of the type \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^n} \] or \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^n} \] using the substitution \(x \rightarrow \frac{\pi}{2}-x\) often simplifies the integral and gives \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^n} = \frac{\pi}{4}. \]

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to evaluate a definite integral with limits from 0 to \( \pi/2 \). The integrand involves a high power of \( \cot(x) \), which suggests that direct integration would be extremely difficult. This is a strong indicator that we should use a property of definite integrals.
Step 2: Key Formula or Approach:
The most useful property for integrals of this type is the "King Property":
\[ \int_{a}^{b} f(x) \,dx = \int_{a}^{b} f(a+b-x) \,dx \] For this specific integral, \( a=0 \) and \( b=\pi/2 \), so the property becomes:
\[ \int_{0}^{\frac{\pi}{2}} f(x) \,dx = \int_{0}^{\frac{\pi}{2}} f\left(\frac{\pi}{2}-x\right) \,dx \] We will also need the trigonometric identity \( \cot(\frac{\pi}{2} - x) = \tan(x) \).
Step 3: Detailed Explanation:
1. Let the integral be I:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \quad \cdots (1) \] 2. Apply the property \( \int_0^a f(x)dx = \int_0^a f(a-x)dx \):
Replace \( x \) with \( \frac{\pi}{2} - x \) in the integrand.
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+\left(\cot\left(\frac{\pi}{2}-x\right)\right)^{101}} \] Using the identity \( \cot(\frac{\pi}{2}-x) = \tan x \), we get:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{101}} \quad \cdots (2) \] 3. Add equation (1) and equation (2):
\[ I + I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1+(\cot x)^{101}} \,dx + \int_{0}^{\frac{\pi}{2}} \frac{1}{1+(\tan x)^{101}} \,dx \] \[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1+\cot^{101}x} + \frac{1}{1+\tan^{101}x} \right) \,dx \] 4. Simplify the integrand:
Let's simplify the expression in the parenthesis. Since \( \tan x = 1/\cot x \):
\[ \frac{1}{1+\cot^{101}x} + \frac{1}{1+\frac{1}{\cot^{101}x}} = \frac{1}{1+\cot^{101}x} + \frac{1}{\frac{\cot^{101}x+1}{\cot^{101}x}} \] \[ = \frac{1}{1+\cot^{101}x} + \frac{\cot^{101}x}{1+\cot^{101}x} = \frac{1+\cot^{101}x}{1+\cot^{101}x} = 1 \] The entire integrand simplifies to 1.
5. Evaluate the final integral:
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \,dx = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \] \[ I = \frac{\pi}{4} \] Step 4: Final Answer:
The value of the integral is \( \frac{\pi}{4} \).