# What is #f(x) = int xe^(2-x) + 3x^2 dx# if #f(0 ) = 1 #?

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The function f(x) = ∫xe^(2-x) + 3x^2 dx with f(0) = 1 evaluates to f(x) = -(x + 2)e^(2-x) + x^3 + C, where C is the constant of integration.

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To find ( f(x) ), we first need to integrate the given function with respect to ( x ), then apply the given condition ( f(0) = 1 ). Integrating ( xe^{2-x} + 3x^2 ) with respect to ( x ) yields:

[ \int xe^{2-x} + 3x^2 , dx = \int xe^{2-x} , dx + \int 3x^2 , dx ]

[ = -xe^{2-x} + \int e^{2-x} , dx + x^3 + C ]

[ = -xe^{2-x} - e^{2-x} + x^3 + C ]

Now, to find ( f(x) ), we'll substitute ( f(0) = 1 ):

[ f(0) = -0e^{2-0} - e^{2-0} + 0^3 + C = -e^2 + C = 1 ]

[ C = 1 + e^2 ]

So, ( f(x) = -xe^{2-x} - e^{2-x} + x^3 + 1 + e^2 ).

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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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