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JEE MainMathematicsDefinite Integration

If I = _ - /2 ^ /2 x^2 x 2x 1 + 2024^ x^3 + x dx = ^2 A - B C , where A, B, C are positive integers and (B,C)=1 , then the value of A + B + C is

Correct answer

65

Step-by-step solution

Let f(x) = x^2 x 2x 1 + 2024^ x^3 + x . Using the property _ -a ^a f(x) dx = ₀^a (f(x) + f(-x)) dx , we get: f(x) + f(-x) = x^2 x 2x 1 + 2024^ x^3 + x + (-x)^2 (-x) (-2x) 1 + 2024^ -x^3 - x = x^2 x 2x ( 1 1 + 2024^ x^3 + x + 2024^ x^3 + x 2024^ x^3 + x + 1 ) = x^2 x 2x So, I = ₀^ /2 x^2 x 2x dx Using the identity x 2x = 1 2 ( 3x + x) , we have: I = 1 2 ₀^ /2 x^2 ( 3x + x) dx Applying Integration by Parts to x^2 kx dx : x^2 kx dx = x^2 kx k - 2x kx k dx = x^2 kx k + 2x kx k^2 - 2 kx k^3 For k=1 : [ x^2 x + 2x x - 2

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