How do you integrate #int x^2e^x dx # using integration by parts?

Answer 1

#intx^2e^xdx = x^2e^x - 2xe^x + 2e^x + c#

The integration by parts formula say

#intudv = uv - intvdu#
So, if we pick #u = x^2# and #dv = e^x# so we'll have #du = 2x# and #v = e^x#
#intx^2e^xdx = x^2e^x - int2xe^xdx#
Now we pick #u = x# and #dv = e^x#, so #v = e^x# and #du = 1#
#intx^2e^xdx = x^2e^x - 2(xe^x - inte^xdx)#

This last integral is tabled, so

#intx^2e^xdx = x^2e^x - 2xe^x + 2e^x + c#
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Answer 2

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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Answer from HIX Tutor

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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