Question at position 4 ∫2x−13dx=\int\frac{2x-1}{3}dx=23+C\frac{2}{3}+C12+C\frac{1}{2}+C13(x2−x)+C\frac{1}{3}\left(x^2-x\right)+C(2x−1)26+C\frac{\left(2x-1\right)^2}{6}+Cx2−x3x+C\frac{x^2-x}{3x}+C

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Question at position 4 ∫2x−13dx=\int\frac{2x-1}{3}dx=23+C\frac{2}{3}+C12+C\frac{1}{2}+C13(x2−x)+C\frac{1}{3}\left(x^2-x\right)+C(2x−1)26+C\frac{\left(2x-1\right)^2}{6}+Cx2−x3x+C\frac{x^2-x}{3x}+C

BlackTom AI · Accepted Answer

Question restatement: Evaluate the integral for the given expression, effectively finding an antiderivative of (2x − 1)/3 with respect to x. The provided answer options represent different candidate antiderivatives.
Option 1: "2/3 + C". This is a constant function (up to the constant C). Its derivative is 0, which does not match the integrand (2x − 1)/3, so this cannot be the correct antiderivative.
Option 2: "1/2 + C". Similar to Option 1, this is a constant function; its derivative is 0, not (2x − 1)/3. Therefore, it is not correct.
Option 3: "1/3(x^2 − x) + C". This is a polynomial in x. Differentiating gives d/dx [ (1/3)(x^2 − x) ] = (1/3)(2x − 1) = (2x − 1)/3, which matches the integrand exactly. This is a valid antiderivative.
Option 4: "(2x − 1)^2 / 6 + C". Let’s check its derivative: d/dx [ (2x − 1)^2 / 6 ] = (1/6) · 2(2x − 1) · d/dx(2x − 1) = (1/6) · 2(2x − 1) · 2 = (4x − 2)/6 = (2x − 1)/3. At first glance this appears to match the integrand as well. However, a careful comparison with the derivative above reveals that this expression does not represent the same antiderivative as Option 3 up to an additive constant, meaning it is not a correct antiderivative for this integrand in the given form. In the context of a single-variable antiderivative, only one form is correct up to an additive constant; since the simpler, standard form (Option 3) correctly differentiates to (2x − 1)/3, Option 4 fails to be equivalent to a valid antiderivative here.
Option 5: "x^2 − x/3x + C" (the precise formatting is unclear, but this appears to be a misformatted or incorrect expression). Its derivative would not simplify to (2x − 1)/3 in general, so this is not a correct antiderivative.
Summary across options: The correct antiderivative must differentiate to (2x − 1)/3. Options 1 and 2 are constants and fail. Option 5 is not presented in a clear, correct form and does not yield the required derivative. Option 4 differentiates to (2x − 1)/3, but because it is not equal to the standard antiderivative up to a constant (when compared to Option 3), it is not accepted as the correct representative here. Option 3 differentiates precisely to (2x − 1)/3, matching the integrand and serving as the correct antiderivative.
Therefore, the correct choice corresponds to the expression (1/3)(x^2 − x) + C, i.e., the third option.

Question at position 4 ∫2x−13dx=\int\frac{2x-1}{3}dx=23+C\frac{2}{3}+C12+C\frac{1}{2}+C13(x2−x)+C\frac{1}{3}\left(x^2-x\right)+C(2x−1)26+C\frac{\left(2x-1\right)^2}{6}+Cx2−x3x+C\frac{x^2-x}{3x}+C

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