MTH1010_10_03_4

Question

MTH1010_10_03_4

BlackTom AI · Accepted Answer

First, consider the general rule: to find the antiderivative, we integrate each term separately and add a constant C at the end.

Option a): x^4 - cos(x) + C
- Integrate 4x^3: ∫4x^3 dx = x^4, since the power rule gives (4/4)x^4 = x^4.
- Integrate sin(x): ∫sin(x) dx = -cos(x).
- Combine: x^4 + (-cos(x)) + C = x^4 - cos(x) + C. This matches the expected result.

Option b): x^4 + cos(x) + C
- Here, sin(x) was incorrectly integrated as +cos(x) instead of -cos(x).
- The correct integral of sin(x) is negative cosine, so this option has the wrong sign for the trigonometric part.

Option c): (4/3) x^4 - cos(x) + C
- The term from integrating 4x^3 is not (4/3)x^4; it is x^4. This option uses an incorrect coefficient, which miscalculates the polynomial part.
- The sin part is correctly integrated as -cos(x), but the x^4 coefficient is wrong.

Option d): (4/3) x^4 + cos(x) + C
- This combines two errors: the coefficient (4/3) is incorrect for the x^4 term, and the sin(x) part has the wrong sign (cos(x) instead of -cos(x)).

Summary: The correct antiderivative is the one that yields x^4 - cos(x) + C, arising from ∫4x^3 dx = x^4 and ∫sin(x) dx = -cos(x). All other options mix up the coefficient or the sign of the cosine term.

MCD2130 - T2 - 2025 11.7 Online Quiz - MCD2130_QUIZ_TEN Differentiation and Integration

MTH1010_10_03_4

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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_08_3

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