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Most Important Selected Qs for JEE AdvancedMathematicsIndefinite Integration

Let e^ x^2 e^x (2 x^2+x+1 ) d x=e^ x^2 f(x)+C where f(x) is some non-zero constant function and C is some arbitrary constant. If the local minimum value of f(x) is equal to m , then: [Note: [ ] denotes greatest integer function.]

Options

  1. Af(x) is increasing in (0, )
  2. Bthe value of _ x 0 (1+f(x))^ 1 / x is equal to 1
  3. Cthe value of ₀^1 (f(x)+e^x ) d x is equal to 2 e .
  4. Dthe value of [ -1 m ] is equal to 2 .

Correct answer

D. the value of [ -1 m ] is equal to 2 .

Step-by-step solution

aligned & x _ I e^ x^2+x (2 x+1) _ II d x+ e^ x^2+x d x & x e^ x^2+x + e^ x^2+x d x aligned f(x)=x e^x aligned & f^ (x)=e^x(1+x)=0 x=-1 & f^ (x)=e^x(2+x) & f^ (-1)=0 Minima aligned f(-1)= -1 e =m [ -1 m ]=[e]=2

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