WEBVTT

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Here is the problem: integral (2x plus 1) cubed dx.

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Find the indefinite integral (antiderivative) of (2 x plus 1) cubed with respect to x.

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Start with the integral.

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Linear substitution u equals ax plus b with the power rule: the integral of u to the power n du equals u to the power n plus 1/(n plus 1), divide by a.

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Simplify and add the constant of integration.

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So the answer is (2 x plus 1) to the power 4 over 8 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.