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COMEDK2023Evening ShiftMathematicsIndefinite IntegrationActual

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

Options

  1. Ax^x+c
  2. Bx x+c
  3. Cx^ x-1 +c
  4. Dx^x x+c

Correct answer

A. x^x+c

Step-by-step solution

Let I = x^x(1 + x) dx . Consider the function f(x) = x^x . To differentiate f(x) , take the natural logarithm on both sides: (f(x)) = (x^x) = x x . Differentiating both sides with respect to x : 1 f(x) f'(x) = d dx (x x) = x 1 x + x 1 = 1 + x . Therefore, f'(x) = f(x)(1 + x) = x^x(1 + x) . Since the derivative of x^x is x^x(1 + x) , the integral of x^x(1 + x) is x^x + c . Answer: x^x+c

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