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

Concept: For definite integrals of the form \[ \int_{0}^{\frac{\pi}{2}} f(\tan x)\,dx \] we use the property \[ \int_{0}^{\frac{\pi}{2}} f(x)\,dx = \int_{0}^{\frac{\pi}{2}} f\!\left(\frac{\pi}{2}-x\right)dx \] Also, \[ \cot\left(\frac{\pi}{2}-x\right) = \tan x \]
Step 1: {Let the given integral be \(I\).} \[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \]
Step 2: {Apply the property \(x \rightarrow \frac{\pi}{2}-x\).} \[ I = \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{101}} \]
Step 3: {Add the two expressions.} \[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1+(\cot x)^{101}} + \frac{1}{1+(\tan x)^{101}} \right)dx \] Let \(t = (\tan x)^{101}\). Then \[ \frac{1}{1+\frac{1}{t}} + \frac{1}{1+t} = 1 \] Thus, \[ 2I = \int_{0}^{\frac{\pi}{2}} 1\,dx \]
Step 4: {Evaluate the integral.} \[ 2I = \frac{\pi}{2} \] \[ I = \frac{\pi}{4} \] Hence, the correct option is (B).