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COMEDK2023Evening ShiftMathematicsIndefinite IntegrationActual

e^x (1+ x+ ^2 x ) d x is equal to

Options

  1. Ae^x x+c
  2. Be^x x+c
  3. Ce^x x+c
  4. De^x x+c

Correct answer

D. e^x x+c

Step-by-step solution

The integral is I = e^x (1 + x + ^2 x) dx . Using the identity 1 + ^2 x = ^2 x , the expression becomes I = e^x ( ^2 x + x) dx . Let f(x) = x . Then f'(x) = ^2 x . The integral is in the form e^x (f(x) + f'(x)) dx , which evaluates to e^x f(x) + c . Therefore, I = e^x x + c . Answer: e^x x+c

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