`f(x) = x(2 - x)^2` Find the most general antiderivative of the function. (Check your answer by differentiation.)

Textbook Question

Chapter 4, 4.9 - Problem 6 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the most general antiderivative, using the following rule, such that:

`int f(x) dx = F(x) + c`

First, you need to open the brackets, such that:

`f(x) = x(2-x)^2 = x*(4 - 4x + x^2) = 4x - 4x^2 + x^3`

`int (4x - 4x^2 + x^3)dx = int 4x dx - int 4x^2 dx + int x^3 dx`

`int (4x - 4x^2 + x^3)dx = 2x^2 - (4/3)x^3 + x^4/4 + c`

Hence, evaluating the most general antiderivative yields `F(x) = 2x^2 - (4/3)x^3 + x^4/4 + c.`

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