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MHT CET202519 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual

x d x (x-1)(x-2) =

Options

  1. A( x-1 x-2 )+c , where c is the constant of integration
  2. B( x-2 (x-1)^2 )+c , where c is the constant of integration
  3. C( x-2 x-1 )+c , where c is the constant of integration
  4. D( (x-2)^2 x-1 )+c , where c is the constant of integration

Correct answer

D. ( (x-2)^2 x-1 )+c , where c is the constant of integration

Step-by-step solution

Decompose the integrand using partial fractions: x (x-1)(x-2) = A x-1 + B x-2 Clearing denominators yields x = A(x-2) + B(x-1) . At x = 1 : 1 = A(-1) , so A = -1 . At x = 2 : 2 = B(1) , so B = 2 . The decomposition becomes -1 x-1 + 2 x-2 . Integrating term by term: - 1 x-1 ,dx + 2 1 x-2 ,dx = - |x-1| + 2 |x-2| + C Applying logarithmic identities: |(x-2)^2| - |x-1| + C = | (x-2)^2 x-1 | + C This expression corresponds to option D. Final answer: D

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