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

the second integral is integrated using the power rule; the first integral is done by integrating by parts#

#IBP formula

which can be simplified as required

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To find ( f(x) = \int xe^{2-x} - 2x^2 , dx ) given ( f(0) = 1 ), integrate the function and then use the given initial condition to solve for the constant of integration. The integral of ( xe^{2-x} - 2x^2 ) with respect to ( x ) is ( -xe^{2-x} + 2x^3/3 ). Then, plug in ( x = 0 ) and ( f(0) = 1 ) to solve for the constant of integration. Solving for the constant yields ( C = 1 + 2/3 ), so ( f(x) = -xe^{2-x} + 2x^3/3 + 5/3 ).

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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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