<p>The value of $\int_{\pi/4}^{\pi/4} \frac{d}{dx}\left(\frac{\cot x}{\cot x + \tan x}\right) dx$ is</p>
Step-by-Step Solution
Key Concept: When the limits of a definite integral are identical (both π/4), the integral equals zero regardless of the integrand, since we're integrating over an interval of zero length.
<p><strong>Step 1:</strong> Identify the limits of integration.</p><p>The integral is: $\int_{\pi/4}^{\pi/4} \frac{d}{dx}\left(\frac{\cot x}{\cot x + \tan x}\right) dx$</p><p>Both the upper and lower limits are π/4.</p><p><strong>Step 2:</strong> Apply the fundamental property of definite integrals.</p><p>By definition, when the upper limit equals the lower limit: $\int_{a}^{a} f(x)\,dx = 0$ for any integrable function f(x).</p><p><strong>Step 3:</strong> Evaluate the integral.</p><p>$\int_{\pi/4}^{\pi/4} \frac{d}{dx}\left(\frac{\cot x}{\cot x + \tan x}\right) dx = 0$</p><p>Note: We could verify that $\frac{\cot x}{\cot x + \tan x} = \frac{\cos^2 x}{1}$ simplifies nicely, but this computation is unnecessary since the limits are identical.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C