Question at position 7 ∫(x3−1x4+2)dx=\int\left(x^3-\frac{1}{x^4}+2\right)dx=x44−13x3+2x+C\frac{x^4}{4}-\frac{1}{3x^3}+2x+Cx44+13x3+2x+C\frac{x^4}{4}+\frac{1}{3x^3}+2x+Cx44−3x3+2x+C\frac{x^4}{4}-\frac{3}{x^3}+2x+C3x2+4x−5+C3x^2+4x^{-5}+C3x2−14x3+C3x^2-\frac{1}{4x^3}+C

Question

Question at position 7 ∫(x3−1x4+2)dx=\int\left(x^3-\frac{1}{x^4}+2\right)dx=x44−13x3+2x+C\frac{x^4}{4}-\frac{1}{3x^3}+2x+Cx44+13x3+2x+C\frac{x^4}{4}+\frac{1}{3x^3}+2x+Cx44−3x3+2x+C\frac{x^4}{4}-\frac{3}{x^3}+2x+C3x2+4x−5+C3x^2+4x^{-5}+C3x2−14x3+C3x^2-\frac{1}{4x^3}+C

BlackTom AI · Accepted Answer

The problem asks us to compute the indefinite integral of the function x^3 − 1/x^4 + 2 with respect to x, and then choose the matching option.
Option 1 appears as: x^4/4 − 1/(3 x^3) + 2x + C. Here, integrating x^3 gives x^4/4, which is correct, but the second term should be +1/(3 x^3) (not −1/(3 x^3)); also the sign for the 1/x^4 term is incorrect since ∫(−x^−4)dx = +1/(3 x^3). Therefore this option misrepresents both the second term and the sign of that term, so it is not correct.
Option 2 appears as: x^4/4 + 1/(3 x^3) + 2x + C (interpreted as + 1/(3 x^3) if read with a denominator that seems intended). While the first and third terms align with the results of ∫x^3dx and ∫2dx, the second term should indeed be +1/(3 x^3) as computed. However, the exact written form here is ambiguous because it shows a +1/3 x^3, which would be (1/3)x^3, not 1/(3 x^3). If we interpret it as intended to mean 1/(3 x^3), this option would match the correct result; if not, it would be incorrect due to missing the reciprocal. Given the typical formatting, there is a risk this is not the intended correct form.
Option 3 appears as: x^4/4 − 3x^3 + 2x + C. This is far from the correct antiderivative: the term −3x^3 would have derivative −9x^2, which does not appear in the original integrand, so this option is inaccurate.
Option 4 appears as: 3x^2 + 4x − 5 + C. This would come from integrating a completely different function; its derivative would be 6x + 4, not our original integrand, so it is plainly incorrect.
Option 5 appears as: 3x^2 − (1/4)x^3 + C. This does not resemble the correct antiderivative either; the x^4 term is missing entirely, and the cubic term’s coefficient and exponent do not align with the integration steps for x^3 and −x^−4.
In summary, each option either miscounts the x^4 term, misrepresents the 1/x^4 term, or omits components entirely. The mathematically correct antiderivative should be x^4/4 + 1/(3 x^3) + 2x + C, obtained by integrating term-by-term: ∫x^3 dx = x^4/4, ∫(−1/x^4) dx = 1/(3 x^3), and ∫2 dx = 2x, plus the constant of integration. The provided options do not cleanly present this exact form, and at least one option introduces a sign error on the 1/(3 x^3) term, while another misplaces a reciprocal altogether. Therefore, none of the options unambiguously matches the correct result, and selecting a correct option would require clarification of the intended formatting for the second term.

Question at position 7 ∫(x3−1x4+2)dx=\int\left(x^3-\frac{1}{x^4}+2\right)dx=x44−13x3+2x+C\frac{x^4}{4}-\frac{1}{3x^3}+2x+Cx44+13x3+2x+C\frac{x^4}{4}+\frac{1}{3x^3}+2x+Cx44−3x3+2x+C\frac{x^4}{4}-\frac{3}{x^3}+2x+C3x2+4x−5+C3x^2+4x^{-5}+C3x2−14x3+C3x^2-\frac{1}{4x^3}+C

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