How do you integrate #int x^2e^x dx # using integration by parts?
The integration by parts formula say
This last integral is tabled, so
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To integrate ( \int x^2 e^x , dx ) using integration by parts, we use the formula ( \int u , dv = uv - \int v , du ). Let's choose ( u = x^2 ) and ( dv = e^x , dx ). Then, ( du = 2x , dx ) and ( v = e^x ).
[ \begin{aligned} \int x^2 e^x , dx &= x^2 e^x - \int 2x e^x , dx \ &= x^2 e^x - 2 \int x e^x , dx \end{aligned} ]
Now, we apply integration by parts to ( \int x e^x , dx ) with ( u = x ) and ( dv = e^x , dx ). Then, ( du = dx ) and ( v = e^x ).
[ \begin{aligned} \int x^2 e^x , dx &= x^2 e^x - 2 (x e^x - \int e^x , dx) \ &= x^2 e^x - 2 (x e^x - e^x) \ &= x^2 e^x - 2x e^x + 2e^x + C \end{aligned} ]
Therefore, ( \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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