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Functions: Number Integers Domain

JEE Maths question with a full step-by-step solution.

Question
The number of integers in the domain of f(x)=1ln⁡cos⁡−1xf(x) = \dfrac{1}{\sqrt{\ln\cos^{-1}x}}.
Solution
Answer: 2
Step 1: Determine when the expression under the square root is positive
ln⁡cos⁡−1x>0  ⟹  cos⁡−1x>e0=1\ln\cos^{-1}x > 0 \implies \cos^{-1}x > e^0 = 1
 Step 2: Find the range of xx Since cos⁡−1x∈[0,π]\cos^{-1}x \in [0, \pi], the condition cos⁡−1x>1\cos^{-1}x > 1 gives:
1<cos⁡−1x≤π  ⟹  −1≤x<cos⁡1≈0.541 < \cos^{-1}x \leq \pi \implies -1 \leq x < \cos 1 \approx 0.54
 Step 3: Count integers in [−1,cos⁡1)[-1, \cos 1) The integers in this interval are −1-1 and 00. Number of integers = 2. Answer: 2
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