How do you integrate #int x^2 e^(-x) dx # using integration by parts?
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To integrate ( \int x^2 e^{-x} , dx ) using integration by parts, follow these steps:
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Identify ( u ) and ( dv ): Let ( u = x^2 ) and ( dv = e^{-x} , dx ).
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Compute ( du ) and ( v ): Calculate ( du ) by taking the derivative of ( u ), which gives ( du = 2x , dx ). Compute ( v ) by integrating ( dv ), which gives ( v = -e^{-x} ).
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Apply the integration by parts formula: [ \int u , dv = uv - \int v , du ]
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Substitute the values: [ \int x^2 e^{-x} , dx = x^2 (-e^{-x}) - \int (-e^{-x}) (2x , dx) ]
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Simplify and integrate: [ = -x^2 e^{-x} + 2 \int x e^{-x} , dx ]
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Repeat integration by parts on the remaining integral: Let ( u = x ) and ( dv = e^{-x} , dx ).
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Compute ( du ) and ( v ): ( du = dx ) and ( v = -e^{-x} ).
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Apply the integration by parts formula again: [ \int x e^{-x} , dx = x (-e^{-x}) - \int (-e^{-x}) , dx ]
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Simplify and integrate: [ = -x e^{-x} + \int e^{-x} , dx ]
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Integrate ( \int e^{-x} , dx ) directly: [ = -x e^{-x} - e^{-x} + C ]
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Combine the results: [ \int x^2 e^{-x} , dx = -x^2 e^{-x} - 2x e^{-x} - 2e^{-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.
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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