The expression [math: ∫04(−x+3x)dx] \int_{0}^{4}{ (-x+3 \sqrt[]{x}) }\: dx

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Begin by parsing the integral: ∫ from 0 to 4 of (-x + 3√x) dx. Break the integral into two simpler parts and integrate each term separately.

First term: ∫_0^4 (-x) dx. The antiderivative of -x is -x^2/2. Evaluating from 0 to 4 gives (-4^2/2) - (-0^2/2) = -16/2 - 0 = -8.

Second term: ∫_0^4 3√x dx = 3 ∫_0^4 x^{1/2} dx. The antiderivative of x^{1/2} is (2/3)x^{3/2}. Multiplying by 3 yields 3*(2/3)x^{3/2} = 2x^{3/2}. Evaluate from 0 to 4: 2*(4)^{3/2} - 2*(0)^{3/2}. Since 4^{1/2} = 2 and 2^3 = 8, we get 2*8 = 16.

Combine the two results: (-8) + 16 = 8.

Now compare with the provided options: a. 20, b. 2, c. 8, d. 16. The value 8 matches option c. Think through potential missteps: a would arise if both terms were miscomputed or if the limits were mishandled; b could come from forgetting the negative sign on -x; d might result from mis-evaluating the square-root term or using 4^{1/2} incorrectly. The key steps are keeping the limits correct and applying the correct antiderivatives for x and x^{1/2} terms, then adding the results.

MUF0092 Mathematics Unit 2 - Semester 2, 2025 8.1 Integration Reminder

The expression [math: ∫04(−x+3x)dx] \int_{0}^{4}{ (-x+3 \sqrt[]{x}) }\: dx

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