Question

Evaluate the definite integral: \[ \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{1}{1 + \sqrt{\cot x}} \, dx \]

Hint:

Whenever a definite integral contains symmetric limits such that \(a+b = \frac{\pi}{2}\) along with combinations of \(\sin x\)/\(\cos x\) or \(\tan x\)/\(\cot x\), the total integral value almost always evaluates simply to \(\frac{b-a}{2}\).

Answer 1

Concept: This problem uses one of the most powerful properties of definite integrals, often referred to as King's Property: \[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \] Additionally, we convert trigonometric functions into terms of sine and cosine using the quotient relationship: \[ \cot x = \frac{\cos x}{\sin x} \] The conversion under King's Property transforms the complementary trigonometric functions (\(\sin \leftrightarrow \cos\)), setting up an algebraic system where adding the original and transformed integrals results in a highly simplified integrand.

Step 1: Express the integral in terms of sine and cosine functions.
Let the given definite integral be denoted by \(I\): \[ I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{1}{1 + \sqrt{\cot x}} \, dx \] We know that \(\sqrt{\cot x} = \frac{\sqrt{\cos x}}{\sqrt{\sin x}}\). Substituting this into the integral expression: \[ I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{1}{1 + \frac{\sqrt{\cos x}}{\sqrt{\sin x}}} \, dx \] Taking a common denominator in the lower fraction and simplifying: \[ I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \quad \cdots (1) \]

Step 2: Calculate the sum of the lower and upper limits.
Let us find the value of \(a + b\): \[ a + b = \frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \] According to King's Property, we can safely replace every instance of \(x\) in the integrand with \(\left(\frac{\pi}{2} - x\right)\).

Step 3: Apply King's Property to generate a secondary equation.
Applying the substitution to equation (1): \[ I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\sin\left(\frac{\pi}{2} - x\right)}}{\sqrt{\sin\left(\frac{\pi}{2} - x\right)} + \sqrt{\cos\left(\frac{\pi}{2} - x\right)}} \, dx \] We know from standard allied angle trigonometric identities that: \[ \sin\left(\frac{\pi}{2} - x\right) = \cos x \quad \text{and} \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x \] Substituting these relationships transforms the integral into: \[ I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \quad \cdots (2) \]

Step 4: Add the original equation (1) and the newly derived equation (2).
By combining both equations, we sum their corresponding expressions under a common integral sign and identical integration limits: \[ I + I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx + \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \] \[ 2I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] The numerator and denominator expressions are identical and cancel out completely, reducing the integrand to 1: \[ 2I = \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} 1 \, dx \]

Step 5: Perform the basic integration and solve for \(I\).
Integrating 1 gives \(x\): \[ 2I = \Big[ x \Big]_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \] Evaluate by inserting the upper and lower boundary values: \[ 2I = \frac{5\pi}{12} - \frac{\pi}{12} \] \[ 2I = \frac{4\pi}{12} = \frac{\pi}{3} \] Dividing across by 2 yields the final integration value: \[ I = \frac{\pi}{6} \] This matches option (B).