How do you integrate #∫(x^3ex^2)/(x^2 +1)^2dx#?

#∫(x^3e^(x^2))/(x^2 +1)^2dx # ?

Answer 1

The answer is #=(e^(x^2))/(2(x^2+1))+C#

Perform the substitution

#u=x^2#, #=>#, #du=2xdx#

The integral is

#I=int(x^3e^(x^2)dx)/(x^2+1)^2#
#=1/2int(ue^udu)/(u+1)^2#

Perform an integration by parts

#intwv'dx=wv-intw'vdx#
#w=ue^u#, #=>#, #w'=ue^u +e^u=e^u(u+1)#
#v'=1/(u+1)^2#, #=>#, #v=-1/(u+1)#

Therefore,

#I=-1/2(ue^u)/(u+1)+1/2inte^udu#
#=e^u/2-(ue^u)/(2(u+1))#
#=e^(x^2)/2-(x^2e^(x^2))/(2(x^2+1))+C#
#=1/2e^(x^2)((x^2+1-x^2))/(x^2+1)+C#
#=(e^(x^2))/(2(x^2+1))+C#
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Answer 2

#int(x^3e^(x^2))/(x^2+1)^2dx=e^(x^2)/(2(x^2+1))+c#

Here,

#I=int(x^3e^(x^2))/(x^2+1)^2dx#
#=int(x^2e^(x^2))/(x^2+1)^2*xdx#
Subst . #color(violet)(x^2=u=>2xdx=du=>xdx=1/2du#

So,

#I=int(ue^u)/(u+1)^2*1/2du#
#=1/2int((u+1-1)e^u)/(u+1)^2du#
#=1/2int{((u+1))/(u+1)^2-1/(u+1)^2}e^udu#

#=1/2{color(blue)(int1/(u+1)e^udu)-color(red) (int1/(u+1)^2e^udu)}#

Integration by Components: in the initial integral

#I#=#1/2{color(blue)([1/(u+1)inte^u-int(-1)/(u+1)^2e^udu])- color(red)(int1/(u+1)^2e^udu)}#
#I#=#1/2{color(brown)(1/(u+1)e^u+color(blue) (int(1)/(u+1)^2e^udu)-color(red)(int1/(u+1)^2e^udu)}#
#I=1/2 *color(brown)(1/(u+1)e^u)+c#
Subst. back #color(violet)( u=x^2# we get
#I=1/2(1/(x^2+1))e^(x^2)+c#

Hence ,

#I=e^(x^2)/(2(x^2+1))+c# .....................................................................................

Note: The question has changed to reflect this.

#x^3ex^2tox^3e^(x^2)#
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Answer 3

To integrate ( \int \frac{x^3 e^{x^2}}{(x^2 + 1)^2} , dx ), you can use the method of substitution. Let ( u = x^2 + 1 ). Then, ( du = 2x , dx ).

After substitution, the integral becomes:

[ \frac{1}{2} \int \frac{e^u}{u^2} , du ]

This integral can be evaluated using integration by parts. Let ( dv = \frac{e^u}{u^2} ), then ( v = -\frac{e^u}{u} ).

Using integration by parts formula:

[ \int u , dv = uv - \int v , du ]

You can find ( \int v , du ), then use this to find the final integral value.

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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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