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Most Important Selected Qs for JEE AdvancedMathematicsIndefinite Integration

If 3 x ^2 x x-3 ^3 x x^4 d x=f(x)+C , where _ x 0 f(x)=1 and C is the constant of integration, then:

Options

  1. Athe value of _ x 0 ₀^x t f(t) d t-2 x^2 1- x =-3
  2. Bthe value of _ x 0 (f(x))^ 1 3 -x^2 x^2 = -1 6
  3. Cif h(x)=x [3] f(x) , then ₀^ h^4(x) d x= 3 8
  4. Dif h(x)=x [3] f(x) , then ₀^ 2 e^ h(x) ( ^2 x- x ) d x=-1

Correct answer

D. if h(x)=x [3] f(x) , then ₀^ 2 e^ h(x) ( ^2 x- x ) d x=-1

Step-by-step solution

aligned & 3 ^2 x x x^3 d x-3 ^3 x x^4 d x & = 3 ^2 x x x^3 d x-3 ( ^3 x ( -1 3 x^3 )- 3 ^2 x x ( -1 3 x^3 ) d x )+C= ^3 x x^3 +C aligned f(x)= ^3 x x^3 (1) _ x 0 ₀^x t ^3 t t^3 d t-2 x^2 1- x = _ x 0 ^3 x x^2 -4 x x = _ x 0 ^3 x x^3 -4 x x =-3 (2) _ x 0 x x -x^2 x^2 = _ x 0 x-x^3 x^3 = D.N.E. (3) h(x)=x x x = x ₀^ ^4 d x= 3 8 (4) ₀^ 2 e^ x ( x x+(- x)) d x= (e^ x x )₀^ / 2 =0-1

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