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JEE Main202116 Mar 2021Evening ShiftMathematicsDefinite IntegrationActual

Consider the integral I = ∫ 0 10 x e x e x - 1 d x where x denotes the greatest integer less than or equal to x . Then the value of I is equal to :

Options

  1. A9 e - 1
  2. B45 e + 1
  3. C45 e - 1
  4. D9 e + 1

Correct answer

C. 45 e - 1

Step-by-step solution

I = ∫ 0 10 x · e x - x + 1 I = ∫ 0 1 0 d x + ∫ 1 2 1 · e 2 - x + ∫ 2 3 2 · e 3 - x + … . . + ∫ 9 10 9 · e 10 - x d x ⇒ I = ∑ n = 0 9 ∫ n n + 1 n · e n + 1 - x d x = - ∑ n = 0 9 n e n + 1 - x n n + 1 = - ∑ n = 0 9 n · e 0 - e 1 = e - 1 ∑ n = 0 9 n = e - 1 · 9 · 10 2 = 45 e - 1

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