# What is #F(x) = int e^(x-2) - 2x^2 dx# if #F(0) = 1 #?

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To find ( F(x) = \int e^{x-2} - 2x^2 , dx ) given ( F(0) = 1 ), first integrate ( e^{x-2} - 2x^2 ) with respect to ( x ) to get ( F(x) ), then solve for the constant of integration using the given initial condition ( F(0) = 1 ).

( F(x) = \int e^{x-2} - 2x^2 , dx )

( F(x) = \int e^{x-2} , dx - \int 2x^2 , dx )

( F(x) = e^{x-2} - \frac{2}{3}x^3 + C )

Now, use the given initial condition ( F(0) = 1 ) to solve for ( C ):

( F(0) = e^{-2} - \frac{2}{3}(0)^3 + C = 1 )

( e^{-2} + C = 1 )

( C = 1 - e^{-2} )

Thus, ( F(x) = e^{x-2} - \frac{2}{3}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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