MTH1010_10_08_3

Question

MTH1010_10_08_3

BlackTom AI · Accepted Answer

Let me restate the problem: find the general antiderivative of 4x^3 + sin(x).

Option a: x^4 - cos(x) + C
- Differentiate to check: d/dx[x^4] = 4x^3, and d/dx[-cos(x)] = sin(x). Summing gives 4x^3 + sin(x), which matches the integrand. This aligns with the power rule for ∫x^n dx and the standard integral ∫sin(x) dx = -cos(x).

Option b: x^4 + cos(x) + C
- Differentiating gives 4x^3 - sin(x). This does not match the original integrand because the sign of sin(x) is wrong.

Option c: (4/3) x^4 - cos(x) + C
- Differentiating yields (16/3) x^3 - sin(x). The coefficient for x^3 is (16/3), which does not equal 4, so this does not match ∫4x^3 dx = x^4, not (4/3)x^4.

Option d: (4/3) x^4 + cos(x) + C
- Differentiating yields (16/3) x^3 - sin(x). Again, the x^3 term coefficient is incorrect and the sign of sin(x) is also inconsistent with the integrand.

Summary: Only option a differentiates to 4x^3 + sin(x), matching the given integrand. The other options fail due to incorrect coefficients or sign errors in the derivative.

MTH1010_10_08_3单项选择题

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Question text Find [math: ∫2x4+3x3−3x+4]\displaystyle\int{2\,x^4+3\,x^3-3\,x+4} [math: dx]. Note: Type "[math: c]" for the integral constant. [input] Check Question 15

MTH1010_10_03_4

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