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Here is the problem: the integral of x(x plus 1) dx.

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Find the indefinite integral (antiderivative) of x(x plus 1) with respect to x.

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Start with the integral.

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Expand into a polynomial.

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Power rule on each term: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

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Simplify and add the constant of integration.

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So the answer is x squared/2 plus x cubed/3 plus C.

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An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

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Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.