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COMEDK2024MathematicsIndefinite IntegrationActual

x( x+2) d x equals to

Options

  1. Ax [1+( x)^2 ]+C
  2. B2 x x+C
  3. Cx(1+ x)^2+C
  4. Dx( x)^2+C

Correct answer

D. x( x)^2+C

Step-by-step solution

Let I = x( x + 2) dx . Expanding the integrand, we get I = (( x)^2 + 2 x) dx . Consider the function f(x) = x( x)^2 . Differentiating f(x) with respect to x using the product rule: f'(x) = d dx [x] ( x)^2 + x d dx [( x)^2] f'(x) = 1 ( x)^2 + x 2 x 1 x f'(x) = ( x)^2 + 2 x . Since the derivative of x( x)^2 is exactly the integrand, the integral is x( x)^2 + C . Answer: x( x)^2+C

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