# How do you integrate #x^3/((x^2+5)^2)#?

We are able to rewrite:

Recognition:

So :

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To integrate ( \frac{x^3}{(x^2 + 5)^2} ), you can use the method of substitution. Let ( u = x^2 + 5 ), then ( du = 2x , dx ).

Now, rewrite the integral in terms of ( u ):

[ \int \frac{x^3}{(x^2 + 5)^2} , dx = \frac{1}{2} \int \frac{1}{u^2} , du ]

This becomes a straightforward integral:

[ \frac{1}{2} \int \frac{1}{u^2} , du = -\frac{1}{2u} + C ]

Now, substitute back for ( u ):

[ -\frac{1}{2(x^2 + 5)} + C ]

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