How to use intergration by parts with this exercise?

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Answer 1 · 2018-01-06T13:33:29

#1#

because one of the limits has infinity we will change it to a limit problem

#int_0^(oo)xe^(-x)dx=lim_(trarroo)int_0^txe^(-x)dx--(1)#

the parts formula

#I=int u(dv)/(dx)dx=uv-intv(du)/(dx)dx#

from #(1)#

#u=x=>(du)/(dx)=1#

#(dv)/(dx)=e^(-x)=>v=-e^(-x)#

#:.I=lim_(trarroo)[-xe^-x-int_0^t(-e^(-x))dx]_0^t#

#:.I=lim_(trarroo)[-xe^-x+int(e^(-x))dx]_0^t#

#:.I=lim_(trarroo)[-xe^-x-e^(-x)]_0^t#

#=lim_(trarroo){(-te^(-t)-e^(-t))-(0-e^0)}#

now #lim_(trarroo)e^(-t)=0#

#:. I=0-0-0+1#

#1#