`int_0^oo x^2e^(-x) dx` Determine whether the integral diverges or converges. Evaluate the integral if it converges.

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We will use integration by parts

`int u dv=uv-int v du`

We will need to apply integration by parts two times in order to eliminate `x^2` from under the integral.

`int_0^infty x^2e^-x dx=|[u=x^2,dv=e^-x dx],[du=2x dx, v=-e^-x]|=`

`-x^2e^-x+2int_0^infty xe^-x dx=|[u=x,dv=e^-x dx],[du=dx,v=-e^-x]|=`

`(-x^2e^-x-2xe^-x)|_0^infty+2int_0^infty e^-x dx=`

`(-x^2e^-x-2xe^-x-2e^-x)|_0^infty=`

`lim_(x to infty)(-x^2e^-x-2xe^-x-2e^-x)-(-0-0-2)=`

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