# How do you integrate #y=xe^(-x)#?

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To integrate ( y = xe^{-x} ), you can use integration by parts.

Let ( u = x ) and ( dv = e^{-x} dx ). Then, ( du = dx ) and ( v = -e^{-x} ).

Applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

[ \int xe^{-x} , dx = -xe^{-x} - \int (-e^{-x}) , dx ]

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

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

So, the integral of ( y = xe^{-x} ) is ( -xe^{-x} - e^{-x} + C ), where ( C ) is the constant of integration.

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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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