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How do you find the antiderivative of #f(x)=(x^3-2x^2+x)/x#?

Answer 1

Emilia Aguilar

#f(x)=(x^3-2x^2+x)/x#

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

#f(x)=(x-1)^2#

#int f(x)=(x-1)^3/3+C#

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Answer 2

Aurora Alvarado

To find the antiderivative of ( f(x) = \frac{x^3 - 2x^2 + x}{x} ), we can simplify the expression first and then integrate.

Simplify the Function:
- Simplify the expression by canceling out the common factor of (x) in the numerator and denominator: [ f(x) = x^2 - 2x + 1 ]
Integrate the Simplified Function:
- To find the antiderivative of (x^2 - 2x + 1), we integrate each term separately: [ \int (x^2 - 2x + 1) , dx = \int x^2 , dx - \int 2x , dx + \int 1 , dx ]
- Applying the power rule of integration, we get: [ \frac{x^3}{3} - \frac{2x^2}{2} + x + C ]
- Simplifying further, we have: [ \frac{x^3}{3} - x^2 + x + C ]

Therefore, the antiderivative of ( f(x) = \frac{x^3 - 2x^2 + x}{x} ) is ( \frac{x^3}{3} - x^2 + x + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.