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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

If f(x) ^3 x 2+ ^2 x d x= | 2-g(x) x |+C , where f(0)= 2 and C is the constant of integration, then: [Note: [k] denotes greatest integer function less than or equal to k .]

Options

  1. A_ x 0 g(x) x^2-x^2 x =2
  2. B₀^ 2 g(x) d x= ⁻¹ ( 1 2 )+ ⁻¹ ( 1 3 )
  3. C_ x 0⁺ [x^2-g(x) ]=0
  4. D₀^ 14 3 g(x) d x= 19 2

Correct answer

D. ₀^ 14 3 g(x) d x= 19 2

Step-by-step solution

( ^2 x-1 ) x ^2 x+2 d x= ( ^2 x x ^2 x+1 - x ^2 x+1 ) d x Put x=t x x d x=d t aligned & ( t t^2+1 - 1 t (t^2+1 ) ) d t= ( t t^2+1 - ( 1 t - t t^2+1 ) ) d t= ( 2 t t^2+1 - 1 t ) d t & = (t^2+1 )- t+C= ( ^2 x+1 x )+C= ( 1+ ^2 x x )+C= ( 2- ^2 x x )+C & g(x)= ^2 x aligned Now, verify the options.

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