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JEE Main2003MathematicsDefinite IntegrationActual

Let f(x) be a function satisfying f^ (x)=f(x) with f(0)=1 and g(x) be a function that satisfies f(x)+g(x)=x^2 . Then the value of the integral ₀^1 f(x) g(x) d x , is

Options

  1. Ae + e ^2 2 + 5 2
  2. Be - e ^2 2 - 5 2
  3. Ce + e ^2 2 - 3 2
  4. De - e ^2 2 - 3 2

Correct answer

D. e - e ^2 2 - 3 2

Step-by-step solution

Let f(x)=e^x ₀^1 f(x) g(x) d x= ₀^1 e^x (x^2-e^x ) d x= ₀^1 x^2 e^x d x- ₀^1 e^ 2 x d x= [ x ^2 e ^ x ]₀^1-2 [ xe ^ x - e ^ x ]₀^1- 1 2 [ e ^ 2 x ]= e - [ e ^2 2 - 1 2 ]-2[ e - e +1]= e - e ^2 2 - 3 2

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