JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

Let f(x) = _2(x^2 + 4) for x 0. To find the inverse function f^ -1 (x), we set y = _2(x^2 + 4) and solve for x: 2^y = x^2 + 4 x^2 = 2^y - 4 x = 2^y - 4 Thus, f^ -1 (x) = 2^x - 4 . The given expression is = _0^ 2 3 f(x)\,dx + _2^4 f^ -1 (x)\,dx. We observe the limits of the first integral and evaluate the function at these points: f(0) = _2(0 + 4) = 2 f(2 3 ) = _2((2 3 )^2 + 4) = _2(12 + 4) = _2(16) = 4 These exactly match the limits of the second integral. Using the standard property of definite integrals for inverse functions: _a^b f(x)\,dx + _ f(a) ^ f(b) f^ -1 (x)\,dx = b f(b) - a f(a) Substituting a = 0 and b = 2 3 : = 2 3 f(2 3 ) - 0 f(0) = 2 3 4 - 0 = 8 3 Therefore, ^2 = (8 3 )^2 = 64 3 = 192. Answer: 192