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JEE Main2014MathematicsIndefinite IntegrationActual

The integral ∫ 1 + x - 1 x e x + 1 x d x , is equal to

Options

  1. Ax + 1 e x + 1 x + c
  2. B- x e x + 1 x + c
  3. Cx - 1 e x + 1 x + c
  4. Dx e x + 1 x + c

Correct answer

D. x e x + 1 x + c

Step-by-step solution

∫ 1 + x - 1 x e x + 1 x d x = ∫ e x + 1 x d x + ∫ x 1 - 1 x 2 e x + 1 x d x Integrating ∫ x ⏟ I . ( 1 − 1 x 2 ) e x + 1 x ⏟ I I d x , by parts I = x e x + 1 x - ∫ e x + 1 x d x + c = ∫ 1 + x - 1 x e x + 1 x d x = ∫ e x + 1 x d x + x e x + 1 x - ∫ e x + 1 x d x + c = x e x + 1 x + c , where c is the constant of integration.

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