# What is #f(x) = int (3x-1)^2-2x+1 dx# if #f(2) = 1 #?

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To find (f(x)) given that (f(2) = 1), you need to integrate the given function and then use the information provided to solve for the constant of integration.

Given (f(x) = \int (3x-1)^2 - 2x + 1 , dx), integrate the function:

[ \int (3x-1)^2 - 2x + 1 , dx = \int (9x^2 - 6x + 1) - 2x + 1 , dx = \int (9x^2 - 8x + 2) , dx ]

Integrate each term separately:

[ \int (9x^2 - 8x + 2) , dx = 3x^3 - 4x^2 + 2x + C ]

Now, given that (f(2) = 1), substitute (x = 2) into the expression:

[ 1 = 3(2)^3 - 4(2)^2 + 2(2) + C ]

[ 1 = 24 - 16 + 4 + C ]

[ 1 = 12 + C ]

[ C = -11 ]

Therefore, (f(x) = 3x^3 - 4x^2 + 2x - 11).

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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.

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.

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.

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