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

##
#∫(x^3e^(x^2))/(x^2 +1)^2dx # ?

The answer is

Perform the substitution

The integral is

Perform an integration by parts

Therefore,

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Here,

So,

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

Integration by Components: in the initial integral

Hence ,

Note: The question has changed to reflect this.

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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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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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