WEBVTT

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Here is the problem: integrate 1/(2x plus 1) dx.

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

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Start with the integral.

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Linear substitution u equals ax plus b: the integral of 1/u du equals ln|u|, divide by a.

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Constant multiple rule: the integral of k times f dx equals k the integral of f dx.

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Simplify and add the constant of integration.

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So the answer is ln(abs(2 x plus 1)) / 2 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.