How do you integrate #int xe^(-x^2)dx# from #[0,1]#?

Answer 1

#int_0^1xe^(-x^2)dx=1/2(1-1/e)#

To determine the definite integral ,we compute the integral itself first then work on the boundaries.

As we know if : #intf(x)=F(x)+C#

Then

#int_a^bf(x)=F(b)-F(a)#
Let us compute the integral #intxe^(-x^2)dx# Let: #color(red)(u(x)=e^(-x^2)#

Then

#color(blue)(du(x)=-2xe^(-x^2)dx# #rArrcolor(blue)(xe^(-x^2)=-1/2du(x)#
#intxe^(-x^2)dx=intcolor(blue)(-1/2du(x))#
#intxe^(-x^2)dx=-1/2intdu(x)#
#intxe^(-x^2)dx=-1/2u(x)+C#
#intxe^(-x^2)dx=-1/2color(red)(e^(-x^2)+C# #" "#

Let us calculate the value of the definite integral:

#int_0^1xe^(-x^2)dx=-1/2(e^(-1^2)-e^(-0^2))#
#int_0^1xe^(-x^2)dx=-1/2(e^(-1)-e^(-0))#
#int_0^1xe^(-x^2)dx=-1/2(e^(-1)-1)#

Therefore,

#int_0^1xe^(-x^2)dx=1/2(1-1/e)#
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Answer 2

To integrate ( \int_0^1 x e^{-x^2} , dx ), you can use the substitution method. Let ( u = -x^2 ). Then ( du = -2x , dx ). Solving for ( dx ), we get ( dx = -\frac{du}{2x} ). Substitute these into the integral and simplify. The integral becomes ( -\frac{1}{2} \int_0^1 e^u , du ). Integrate ( e^u ) with respect to ( u ) from ( 0 ) to ( 1 ). Then evaluate the expression at the upper limit and subtract the value at the lower limit. This gives ( -\frac{1}{2} (e^1 - e^0) ), which simplifies to ( -\frac{1}{2} (e - 1) ). Therefore, ( \int_0^1 x e^{-x^2} , dx = -\frac{1}{2} (e - 1) ).

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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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