`int_(-oo)^0 xe^(-4x) 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 udv=uv-int vdu`

`int_-infty^0 xe^(-4x)dx=|[u=x,dv=e^(-4x)dx],[du=dx,v=-1/4e^(-4x)]|=`

`-1/4xe^(-4x)|_-infty^0+1/4int_-infty^0 e^(-4x)dx=`

`(-1/4xe^(-4x)-1/16e^(-4x))|_-infty^0=`

`-1/4cdot0cdote^0-1/16e^0+lim_(x to -infty)[e^(-4x)(1/4x+1/16)]=`

`0-1/16+infty(-infty+1/16)=-infty`  

As we can see the integral diverges.                                 

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We will use integration by parts

`int udv=uv-int vdu`

`int_-infty^0 xe^(-4x)dx=|[u=x,dv=e^(-4x)dx],[du=dx,v=-1/4e^(-4x)]|=`

`-1/4xe^(-4x)|_-infty^0+1/4int_-infty^0 e^(-4x)dx=`

`(-1/4xe^(-4x)-1/16e^(-4x))|_-infty^0=`

`-1/4cdot0cdote^0-1/16e^0+lim_(x to -infty)[e^(-4x)(1/4x+1/16)]=`

`0-1/16+infty(-infty+1/16)=-infty`  

As we can see the integral diverges.                                 

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