If $\alpha=\int_{0}^{2 \sqrt{3}} \log _{2}\left(x^{2}+4\right) dx+\int_{2}^{4} \sqrt{2^{x}-4} dx$, then $\alpha^{2}$ is equal to ____.

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Definite Integrals

α = \int_{0}^{23} lo g_{2} (x^{2} + 4) d x + \int_{2}^{4} 2^{x} - 4 d x

, then

α^{2}

is equal to ____.

Answer: 192

Let

f (x) = lo g_{2} (x^{2} + 4)

for

x \geq 0

; then

f (0) = lo g_{2} 4 = 2

and

f (23) = lo g_{2} 16 = 4

. Its inverse satisfies

x = 2^{y} - 4

, so

\int_{2}^{4} 2^{x} - 4 d x = \int_{f (0)}^{f (23)} f^{- 1} (x) d x

. By the inverse-function area relation

\int_{a}^{b} f (x) d x + \int_{f (a)}^{f (b)} f^{- 1} (x) d x = b f (b) - a f (a)

, the sum equals

23 \cdot 4 - 0 \cdot 2 = 83

. Hence

α = 83

and

α^{2} = 192

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