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COMEDK2024Evening ShiftMathematicsIndefinite IntegrationActual

The value of d x 2 x-x^2 is

Options

  1. A⁻¹(2 x-1)+C
  2. B⁻¹(x+1)+C
  3. C- 2 x-x^2 +C
  4. D⁻¹(x-1)+C

Correct answer

D. ⁻¹(x-1)+C

Step-by-step solution

The integral is I = dx 2x - x^2 . Complete the square for the expression inside the square root: 2x - x^2 = -(x^2 - 2x) = -(x^2 - 2x + 1 - 1) = 1 - (x - 1)^2 . Substituting this into the integral: I = dx 1 - (x - 1)^2 . Using the standard integral formula du a^2 - u^2 = ⁻¹ ( u a ) + C , where u = x - 1 and a = 1 : I = ⁻¹ ( x - 1 1 ) + C = ⁻¹(x - 1) + C . Answer: ⁻¹(x-1)+C

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