How do you use integration by parts to find #intxe^-x dx#?

Answer 1
#=-e^-x(1+x)+c#, where c is a constant

Explanation

Using Integration by Parts,

#int(I)(II)dx=(I)int(II)dx-int((I)'int(II)dx)dx#
where #(I)# and #(II)# are functions of #x#, and #(I)# represents which will be differentiated and #(II)# will be integrated subsequently in the above formula

Similarly following for the problem,

#=x*inte^-xdx-int((x)'inte^-xdx)dx#
#=x*e^-x/(-1)+inte^-xdx#
#=-x*e^-x+e^-x/(-1)+c#, where c is a constant
#=-x*e^-x-e^-x+c#, where c is a constant
#=-e^-x(1+x)+c#, where c is a constant
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Answer 2

To integrate ( x e^{-x} ) using integration by parts, you would typically follow these steps:

  1. Choose which part of the integrand to designate as ( u ) and which part to designate as ( dv ) in the integration by parts formula: ( \int u , dv = uv - \int v , du ).

  2. Differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ).

  3. Substitute the expressions for ( u ), ( du ), ( v ), and ( dv ) into the integration by parts formula.

  4. Simplify and evaluate the resulting integral.

For ( \int x e^{-x} , dx ), a common choice is to let ( u = x ) and ( dv = e^{-x} , dx ).

Then, ( du = dx ) and ( v = -e^{-x} ).

Substituting into the integration by parts formula:

( \int x e^{-x} , dx = -xe^{-x} - \int (-e^{-x}) , dx )

( = -xe^{-x} + \int e^{-x} , dx )

( = -xe^{-x} - e^{-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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