Question text a) Find [math: ∫3x3+5x+5]\displaystyle\int{3\,x^3+5\,x+5} [math: dx] .
Note: Type [math: c] for the integral constant. [input] b) Hence, find the equation of the function that has [math: f′(x)=3x3+5x+5]\displaystyle {f}'\left( x \right)={3\,x^3+5\,x+5} and passes through the point [math: (1,4)]({1},{4}) [math: f(x)=][input] Check Question 16

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The task has two parts: a) compute an indefinite integral, b) use a point to determine the constant of integration for the antiderivative.

Option 1: 0.75x^4 + 2.5x^2 + 5x + c
This is the correct form for the antiderivative of 3x^3 + 5x + 5, since integrating term by term yields ∫3x^3 dx = (3/4)x^4 = 0.75x^4, ∫5x dx = (5/2)x^2 = 2.5x^2, ∫5 dx = 5x, and the constant of integration is typically written as c. The result matches the standard antiderivative structure, with the constant unspecified.

Option 2: 0.75x^4 + 2.5x^2 + 5x - 4.25
This expression corresponds to choosing a specific constant C = -4.25 in the antiderivative f(x) = (3/4)x^4 + (5/2)x^2 + 5x + C. To determine C, plug the given point (1,4) into f: f(1) = 0.75 + 2.5 + 5 + C = 8.25 + C. Setting f(1) = 4 gives C = 4 − 8.25 = -4.25. Hence this form represents the particular antiderivative that passes through (1,4).

In part (a), the presence of the + c is standard for indefinite integrals; in part (b), substituting the point selects the specific constant to satisfy the condition f(1) = 4, resulting in the constant -4.25. The numbers 0.75 and 2.5 arise directly from the coefficients during integration: 3/4 and 5/2, respectively, and 5x comes from ∫5 dx = 5x.

MCD2130 - T2 - 2025 MCD2130 Test 3

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