JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

Let I = _ 0 ^ 2 x(x^2 + x + 1) x + 1 \, x^4 + x^2 + 1 \, dx. Using the identity x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1): I = _ 0 ^ 2 x \, x^2 + x + 1 x + 1 \, (x^2 + x + 1)(x^2 - x + 1) \, dx Cancelling x^2 + x + 1 : I = _ 0 ^ 2 x (x + 1)(x^2 - x + 1) \, dx Using (x + 1)(x^2 - x + 1) = x^3 + 1: I = _ 0 ^ 2 x x^3 + 1 \, dx Substitution: Let t = x^ 3/2 , so dt = 3 2 x^ 1/2 \, dx x \, dx = 2 3 \, dt. Also, t^2 = x^3, so x^3 + 1 = t^2 + 1. Changing limits: x = 0 t = 0 x = 2 t = 2^ 3/2 = 2 2 Therefore: I = 2 3 _ 0 ^ 2 2 dt t^2 + 1 Using dt t^2 + a^2 = _e |t + t^2 + a^2 | + C: I = 2 3 [ _e (t + t^2 + 1 ) ]_ 0 ^ 2 2 I = 2 3 [ _e (2 2 + 8 + 1 ) - _e(1) ] I = 2 3 _e (2 2 + 3 ) I = 2 3 _e (3 + 2 2 ) Hence, the correct option is (3)\ 2 3 _e (3 + 2 2 ).