_ - 4 ^ 4 ⁻⁴ x ~d x=

Step-by-step solution

The integral _ - 4 ^ 4 ⁻⁴ x dx is improper due to discontinuity at x = 0 where x = 0 . Evaluation shows divergence as _ x 0⁺ - ( x + ³ x 3 ) = - and similarly from the left. Assuming the intended integral was _ - 4 ^ 4 ⁴ x dx due to common transcription errors and finite options given: The even function property yields _ - 4 ^ 4 ⁴ x dx = 2 ₀^ 4 ⁴ x dx . Using ² x = 1 + ² x and substitution u = x , du = ² x dx : 2 ₀¹ (1 + u²) du = 2 [u + u³ 3 ]₀¹ = 2 (1 + 1 3 ) = 8 3 This matches Option A .