Question

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

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{1}{2} \)
• D. 1

Hint:

For definite integrals involving expressions like \( \cot x \) or \( \tan x \), use the property \( \int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx \) to simplify the integration.

Answer 1

The Correct Option is A

The following calculation utilizes the property of definite integrals: \[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \] Consider the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1 + (\cot x)^{101}} \] Applying the aforementioned property: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1 + (\cot (\frac{\pi}{2} - x))^{101}} = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1 + (\tan x)^{101}} \] Summing the two expressions for I: \[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1 + (\cot x)^{101}} + \frac{1}{1 + (\tan x)^{101}} \right) dx \] Using the identity: \[ \frac{1}{1 + a^n} + \frac{1}{1 + \frac{1}{a^n}} = 1 \quad \text{for } a>0 \] This implies: \[ \frac{1}{1 + (\cot x)^{101}} + \frac{1}{1 + (\tan x)^{101}} = 1 \] Therefore: \[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2} \] Which yields: \[ I = \frac{\pi}{4} \]