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AP EAMCET202119 Aug 2021Morning ShiftMathematicsIndefinite IntegrationActual

If x 4 ( x - 1 ) ( x - 2 ) = f ( x ) + A x - 1 + B x - 2 , then

Options

  1. Af ( x ) = x 2 − 3 x + 7
  2. Bf ( x ) = x 2 + 3 x + 7
  3. CA + B = 17
  4. DA − B = − 18

Correct answer

B. f ( x ) = x 2 + 3 x + 7

Step-by-step solution

Given x 4 ( x - 1 ) ( x - 2 ) = f ( x ) + A x - 1 + B x - 2 ⇒ x 4 ( x - 1 ) ( x - 2 ) = f ( x ) x - 1 x - 2 + A x - 2 + B x - 1 x - 1 x - 2 ⇒ x 4 = f ( x ) x - 1 x - 2 + A x - 2 + B x - 1 On putting x = 1 , we get A = - 1 On putting x = 2 , we get B = 16 Then, x 4 = f ( x ) x - 1 x - 2 - x - 2 + 16 x - 1 ⇒ x 4 - 16 x + 16 + x - 2 = f ( x ) x - 1 x - 2 ⇒ f ( x ) = x 4 - 15 x + 14 x - 1 x - 2 On adding and Subtracting x 2 , we get f ( x ) = x 4 - 15 x + 14 + x 2 - x 2 x - 1 x - 2 = x 4 - x 2 -

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