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Here is the problem: integrate x sqrt(x squared plus 1).

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

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Start with the integral.

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Substitute u equals x squared plus 1, so du equals (2x) dx.

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the integral of the square root of u du equals (2 over 3)u to the power 3 over 2).

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Substitute back u equals x squared plus 1.

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Simplify and add the constant of integration.

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So the answer is (x squared plus 1) to the power 3 over 2) / 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.