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JEE Main202523 Jan 2025Evening ShiftMathematicsDefinite IntegrationActual

If I = ₀^ 2 ^ 3 2 x ^ 3 2 x+ ^ 3 2 x ~d x , then ₀²¹ x x x ^4 x+ ^4 x ~d x equals :

Options

  1. A^2 12
  2. B^2 4
  3. C^2 16
  4. D^2 8

Correct answer

C. ^2 16

Step-by-step solution

aligned & I= ₀^ 2 ( x)^ 3 2 d x ( x)^ 3 2 x+( x)^ 3 2 = ₀^ 2 ^ 3 2 ( 2 -x ) d x ^ 3 2 ( 2 -x )+ ^ 3 2 ( 2 -x ) & Adding 2 l= ₀^ 2 ( x)^ 3 2 +( x)^ 3 2 ( x)^ 3 2 +( x)^ 3 2 d x= 2 & I₀= ₀^ 2 x x x ^4 x+ ^4 x d x= ₀^ 2 ( 2 -x ) x x ( x)^4+( x)^4 d x aligned Adding, 2 I₀= ₀^ 2 2 ( x) x ( ^4 x+ ^4 x ) d x I₀= 4 ₀^ 2 x ( ^2 x ) d x 1+ ^4 x put ^2 x=t 2 x ^2 x d x=d t aligned & I₀= 4 ₀^ d t 2 (1+t^2 ) = . 8 ( ⁻¹ t ) |₀ ^ = 8 ( 2 -0 ) & I₀= ^2 16 aligned

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