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JEE Main202522 Jan 2025Morning ShiftMathematicsDefinite IntegrationActual

Let for f(x)=7 ^8 x+7 ^6 x-3 ^4 x-3 ^2 x, I ₁= ₀^ / 4 f(x) d x and I ₂= ₀^ / 4 x f(x) d x . Then 7 I ₁+12 I ₂ is equal to :

Options

  1. A2
  2. B1
  3. C2

Correct answer

B. 1

Step-by-step solution

aligned f(x) & =7 ^8 x+7 ^6 x-3 ^4 x-3 ^2 x & =7 ^6 x (1+ ^2 x )-3 ^2 x (1+ ^2 x ) & = (7 ^6 x-3 ^2 x ) (1+ ^2 x ) & = (7 ^6 x-3 ^2 x ) ^2 x l₁= & ₀^ 4 f(x) d x= ₀^ 4 (7 ^6 x-3 ^2 x ) ^2 x d x & = . ( 7 ^7 x 7 - 3 ^3 x 3 ) |₀ ^ 4 =1-1=0 aligned aligned I₂= & ₀^ 4 x f(x) d x= ₀^ 4 x (7 ^6 x-3 ^2 x ) ^2 x d x & = .x ( ^7 x- ^3 x ) |₀ ^ 4 - ₀^ 4 1 ( ^7 x- ^3 x ) d x & =0- ₀^ 4 ^3 x ( ^2 x-1 ) ( ^2 x+1 ) d x & = ₀^ 4 ( ^3 x- ^5 x ) ^2 x d x= ^4 x 4 - . ^6 x 6 |₀ ^ 4 & = 1 12 aligned Hence 7 I₁+12 I₂=1

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