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AP EAMCET202316 May 2023Morning ShiftMathematicsDefinite IntegrationActual

If ₀^ 2024 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x=k , then ( 2 k +1 )=

Options

  1. A2023
  2. B2025
  3. C2022
  4. D2024

Correct answer

B. 2025

Step-by-step solution

Given : ₀^ 2024 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x=k ....(i) Let I= ₀^ 2024 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x ...(ii) aligned & I=2024 ₀^ 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x & I=2024 2 ₀^ / 2 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x & I=2024 2 ₀^ / 2 2023^ ^2 ( 2 -x ) 2023^ ^2 ( 2 -x ) +2023^ ^2 ( 2 -x ) d x aligned I=2024 2 ₀^ 2 2023^ ^2 x 2023^ ^2 x +2023^ ^2 x d x ...(iii) Adding (ii) and (iii) : 2 I=2024 2 ₀^ / 2 1 d x 2 I=2024 2 2 I=1012 From eqns. (i) and (ii) : k= I k=1012 Now, 2 k +1= 2 1012 +1=2024+1

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