`f(x) = (x + 1)(2x - 1)` Find the most general antiderivative of the function. (Check your answer by differentiation.)

Textbook Question

Chapter 4, 4.9 - Problem 5 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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

`f(x) = (x+1)(2x-1) = 2x^2 + x - 1`

You need to evaluate the most antiderivative of the function, such that:

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

`int (2x^2 + x - 1) dx = int 2x^2 dx + int xdx - int dx`

`int (2x^2 + x - 1) dx =(2/3)x^3 + (1/2)x^2 - x + c`

Hence, evaluating the most antiderivative of the function yields `F(x) = (2/3)x^3 + (1/2)x^2 - x + c.`

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Seema Adiga | High School Teacher | (Level 2) Adjunct Educator

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`f(x) = (x+1)(2x-1) `

First you have open the brackets by multiplying the given binomoals

`f(x)= 2x^2+x-1`

Now evaluate the antiderivation (integration)

`int f(x)dx = int (2x^2+x-1)`

         

`= int 2x^2 + int x -int 1 `

         

`= 2 *(x^3)/3+ (x^2)/2 - x + C `

           = `2/3 x^3 + 1/2 x^2 - x+ c`

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