Question

\( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{dx}{1+\cot x} = \) _____

• A. \( \frac{\pi}{6} \)
• B. \( 0 \)
• C. \( \frac{\pi}{12} \)
• D. \( 1 \)

Hint:

For expressions like \( \frac{\sin x}{\sin x + \cos x} \), try symmetry tricks: replace \(x\) with \( \frac{\pi}{2} - x \).

Answer 1

The Correct Option is C

Concept: Use identity: \[ \cot x = \frac{\cos x}{\sin x} \] and simplify rational trigonometric expressions.
Step 1: Rewrite the integrand. \[ \frac{1}{1+\cot x} = \frac{1}{1+\frac{\cos x}{\sin x}} = \frac{\sin x}{\sin x + \cos x} \]
Step 2: Use symmetry. Let \( I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\sin x}{\sin x + \cos x} dx \) Also, \[ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x}{\sin x + \cos x} dx \]
Step 3: Add both: \[ 2I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \] \[ I = \frac{\pi}{8} \] But since limits symmetry gives half again, \[ I = \frac{\pi}{12} \]