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JEE Main202623 January 2026Evening ShiftMathematicsDefinite IntegrationActual

The number of elements in the set S = x: x [0,100] . and . ₀^ x t² (x-t) d t=x² is _ _ _ _ .

Correct answer

16

Step-by-step solution

Using u = x - t : ₀^x t^2 (x-t) ,dt = ₀^x (x-u)^2 u ,du . Expanding and integrating term by term: = x^2(1- x) - 2x(-x x + x) + (-x^2 x + 2x x + 2 x - 2) = x^2 + 2 x - 2 . Setting equal to x^2 : 2 x - 2 = 0 x = 1 x = 2n . In [0, 100] : 2n 100 n 50 15.9 . So n = 0, 1, 2, , 15 , giving 16 elements.

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