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.

Questions

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

Single choice

MTH1010_10_08_3

Options

A.a. a)

B.b. b)

C.c. c)

D.d. d)

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Step-by-Step Analysis

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),......Login to view full explanation

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