JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx\), then \(\alpha^2\) is equal to _______.
- A 190
- B 191
- C 192
- D 193
Answer & Solution
Correct Answer
(C) 192
Step-by-step Solution
Detailed explanation
Let \(f(x) = \log_2(x^2 + 4)\) for \(x \ge 0\). To find the inverse function \(f^{-1}(x)\), we set \(y = \log_2(x^2 + 4)\) and solve for \(x\): \(2^y = x^2 + 4\) \(x^2 = 2^y - 4\) \(x = \sqrt{2^y - 4}\) Thus, \(f^{-1}(x) = \sqrt{2^x - 4}\). The given expression is…
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