How do you integrate #int x^2 e^(x) dx # using integration by parts?
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate ( \int x^2 e^{x} , dx ) using integration by parts, follow these steps:

Identify ( u ) and ( dv ): Let ( u = x^2 ) and ( dv = e^{x} , dx ).

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} ).

Apply the integration by parts formula: [ \int u , dv = uv  \int v , du ]

Substitute the values: [ \int x^2 e^{x} , dx = x^2 (e^{x})  \int (e^{x}) (2x , dx) ]

Simplify and integrate: [ = x^2 e^{x} + 2 \int x e^{x} , dx ]

Repeat integration by parts on the remaining integral: Let ( u = x ) and ( dv = e^{x} , dx ).

Compute ( du ) and ( v ): ( du = dx ) and ( v = e^{x} ).

Apply the integration by parts formula again: [ \int x e^{x} , dx = x (e^{x})  \int (e^{x}) , dx ]

Simplify and integrate: [ = x e^{x} + \int e^{x} , dx ]

Integrate ( \int e^{x} , dx ) directly: [ = x e^{x}  e^{x} + C ]

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.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7