<p>Evaluate the following: <br> 21. \(\int_0^{1/n} \tan^{-1}\sqrt{Vx-1} \, dx\)</p>
Step-by-Step Solution
Key Concept: Recognize that √(nx-1) is only real for x ≥ 1/n, making the integrand zero on [0, 1/n). The integral evaluates to 0 because the domain of definition is a single point.
<p><strong>Step 1:</strong> Analyze the domain of the integrand f(x) = tan⁻¹√(nx-1)</p><p>For the square root to be defined in ℝ, we need: nx - 1 ≥ 0, which gives x ≥ 1/n</p><p><strong>Step 2:</strong> Check the integration interval [0, 1/n]</p><p>For x ∈ [0, 1/n), we have nx < 1, so nx - 1 < 0, making √(nx-1) undefined.</p><p>At x = 1/n, we have √(n·(1/n) - 1) = √0 = 0, so f(1/n) = tan⁻¹(0) = 0</p><p><strong>Step 3:</strong> Evaluate the integral</p><p>Since the integrand is undefined (hence effectively zero) on [0, 1/n) and equals 0 at the endpoint x = 1/n, the integral over a single point is zero.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0