How would you integrate #(x^2)(e^(x-1))#?

Answer 1

Here, you would have to perform some partial integration.

lets first set #f(x) = x^2# and #g'(x) = e^(x-1)#
and use #u = f(x)# and #v = g(x)#
then #du = f'(x) dx# and #dv = g'(x)#

and the integral by parts formula is applied:

#intudv = uv - intv du#

Let's now obtain the values:

#u = x^2# and #v= e^(x-1)#
#du = 2x# and #dv= e^(x-1)#

Thus:

#int x^2e^(x-1) dx = x^2e^(x-1) - int 2xe^(x-1) dx#
#int x^2e^(x-1) dx = x^2e^(x-1) - 2int xe^(x-1) dx#
now lets solve for #color(red)(int xe^(x-1) dx)#
#color(red)(u = x)# and #color(red)(v=e^(x-1))#
#color(red)(du = 1)# and #color(red)(dv=e^(x-1))#

Thus:

#color(red)( int xe^(x-1) = xe^(x-1) - int 1e^(x-1)#

which is equivalent to:

#color(red)( int xe^(x-1) = xe^(x-1) - e^(x-1)#

We can now reintroduce that into our original issue and obtain:

#int x^2e^(x-1) dx = x^2e^(x-1) - 2int xe^(x-1) dx#
is equal to, #int x^2e^(x-1) dx = x^2e^(x-1) - 2(xe^(x-1) - e^(x-1))#

where we can reduce things to:

#int x^2e^(x-1) = e^(x-1)(x^2 - 2x + 2)#
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Answer 2

To integrate ( (x^2)(e^{x-1}) ), you can use integration by parts. Let ( u = x^2 ) and ( dv = e^{x-1}dx ). Then, ( du = 2x dx ) and ( v = e^{x-1} ). Applying the integration by parts formula:

[ \int (x^2)(e^{x-1}) dx = x^2 e^{x-1} - \int 2x e^{x-1} dx ]

Now, integrate ( 2x e^{x-1} ) using integration by parts again:

Let ( u = 2x ) and ( dv = e^{x-1}dx ). Then, ( du = 2 dx ) and ( v = e^{x-1} ).

[ \int 2x e^{x-1} dx = 2x e^{x-1} - \int 2 e^{x-1} dx ]

[ = 2x e^{x-1} - 2e^{x-1} + C ]

Substitute this back into the original equation:

[ \int (x^2)(e^{x-1}) dx = x^2 e^{x-1} - (2x e^{x-1} - 2e^{x-1}) + C ]

[ = x^2 e^{x-1} - 2x e^{x-1} + 2e^{x-1} + C ]

Therefore, the integral of ( (x^2)(e^{x-1}) ) is ( x^2 e^{x-1} - 2x e^{x-1} + 2e^{x-1} + 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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