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

Question

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

BlackTom AI · Accepted Answer

Let's break down the given integral step by step and then evaluate each proposed antiderivative.
Option 1: x^(4)/4 − 3x^3 + 2x + C. This expression would have an x^3 term with a coefficient of −3, which does not align with the original integrand components. The integral of x^3 is x^4/4, but there is no reason to introduce a −3x^3 term from integrating x^3 or −x^4, so this option adds an extraneous and incorrect term.
Option 2: x^(4)/4 + 1/(3x^3) + 2x + C. This matches the standard antiderivative step by step: ∫x^3 dx = x^4/4, ∫(−x^−4) dx = ∫−x^−4 dx = −(x^−3)/−3 = 1/(3x^3), and ∫2 dx = 2x. Combining these yields x^4/4 + 1/(3x^3) + 2x + C, which is consistent with the integration rules and the original integrand.
Option 3: 3x^2 + 4x − 5 + C. This option resembles a polynomial with x^2 and x terms, which would arise if the integrand were a polynomial like x^3 or similar, not from x^3 − x^4 + 2 or from −x^4. The coefficients and terms do not correspond to the given integrand’s components, so this is incorrect.
Option 4: 3x^2 − 1/4x^3 + C. Here we see a combination that could come from integrating pieces like 3x or −(1/4)x^3, but it does not align with the integrand x^3 − x^4 + 2. Specifically, the presence of −(1/4)x^3 would come from integrating −(1/4)x^2, which is not present in the original expression, making this option invalid.
Option 5: x^(4)/4 − 1/(3x^3) + 2x + C. This option is very close in structure to the correct antiderivative, but the sign of the middle term is incorrect: ∫(−x^−4) dx yields +1/(3x^3), not −1/(3x^3). Therefore this one mismatches the proper sign for that component.
In summary, the only option that faithfully applies the antiderivative rules to each piece of the integrand is the one that yields x^4/4 plus 1/(3x^3) plus 2x, with the constant C. The other choices introduce extraneous terms, incorrect coefficients, or wrong signs that do not reflect the original integrand.

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

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