How do you find the antiderivative of #f(x)=(x^32x^2+x)/x#?
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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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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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