Question

Evaluate : \( \int_{\pi/12}^{5\pi/12} \frac{dx}{1 + \sqrt{\cot x}} \)

Hint:

When limits add up to \( \pi/2 \) and the integrand involves \( \sin/ \cos \) or \( \tan/ \cot \), the property usually leads to a constant integrand.

Answer 1

Concept:
• Definite integral property: \( \int_a^b f(x) dx = \int_a^b f(a + b - x) dx \).

Step 1: Simplify the integrand
\[ I = \int_{\pi/12}^{5\pi/12} \frac{dx}{1 + \frac{\sqrt{\cos x}}{\sqrt{\sin x}}} = \int_{\pi/12}^{5\pi/12} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx \quad \dots(1) \]

Step 2: Apply the integral property
Here \( a + b = \frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \).
Replacing \( x \) by \( \pi/2 - x \):
\[ I = \int_{\pi/12}^{5\pi/12} \frac{\sqrt{\sin(\pi/2 - x)}}{\sqrt{\sin(\pi/2 - x)} + \sqrt{\cos(\pi/2 - x)}} dx \]
\[ I = \int_{\pi/12}^{5\pi/12} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx \quad \dots(2) \]

Step 3: Add equations (1) and (2)
\[ 2I = \int_{\pi/12}^{5\pi/12} \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx = \int_{\pi/12}^{5\pi/12} 1 \, dx \]
\[ 2I = [x]_{\pi/12}^{5\pi/12} = \frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \]

Step 4: Final value
\[ I = \frac{\pi}{6} \]