JEE Main 2026 Apr 2 Shift 1, Mathematics Q25: Some Properties of Definite Integrals

JEE Main2026Apr 2, Shift 1Mathematics

If $α = \int_{0}^{23} lo g_{2} (x^{2} + 4) d x + \int_{2}^{4} 2^{x} - 4 d x$ , then $α^{2}$ is equal to ________.

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Solution

Let $f (x) = lo g_{2} (x^{2} + 4)$ . Then $f$ is increasing on $[0, \infty)$ with $f (0) = 2$ and $f (23) = lo g_{2} (12 + 4) = 4$ . Its inverse function is found by solving $y = lo g_{2} (x^{2} + 4)$ : $x^{2} = 2^{y} - 4$ , so $f^{- 1} (y) = 2^{y} - 4$ for $y \in [2, 4]$ .

The two integrals are exactly the area under $f$ on $[0, 23]$ and the area under $f^{- 1}$ on $[2, 4]$ . By the inverse-function area identity,

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

$α = 23 \cdot 4 - 0 \cdot 2 = 83$

$∴ α^{2} = 64 \cdot 3 = 192$ .

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JEE Main 2026 Apr 2 Shift 1, Mathematics Q25: Some Properties of Definite Integrals

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