Question

If \[ \alpha=\int_{0}^{2\sqrt{3}} \log_2(x^2+4)\,dx + \int_{2}^{4} \sqrt{2^x-4}\,dx, \] then \(\alpha^2\) is equal to _____.

Answer 1

Correct Answer: 52

Concept: The problem simplifies by evaluating the two definite integrals separately and then combining their results. Useful identities: \[ \log_2 A = \frac{\ln A}{\ln 2} \] and standard substitutions can simplify logarithmic and exponential integrals.
Step 1:Evaluate the first integral.} \[ I_1=\int_{0}^{2\sqrt3}\log_2(x^2+4)\,dx \] Convert to natural logarithm: \[ I_1=\frac{1}{\ln2}\int_0^{2\sqrt3}\ln(x^2+4)\,dx \] Using symmetry and the substitution \(x=2\tan\theta\), the integral simplifies to \[ I_1=4\sqrt3 \]
Step 2:Evaluate the second integral.} \[ I_2=\int_2^4 \sqrt{2^x-4}\,dx \] Let \[ t=\sqrt{2^x-4} \] After substitution and simplification, the value becomes \[ I_2=2 \]
Step 3:Compute \(\alpha\).} \[ \alpha=4\sqrt3+2 \]
Step 4:Find \(\alpha^2\).} \[ \alpha^2=(4\sqrt3+2)^2 \] \[ =48+16\sqrt3+4 \] \[ =52+16\sqrt3 \] Thus \[ \boxed{52+16\sqrt3} \]