How do you evaluate the definite integral #int_0^(oo)x^3e^(-x)dx#?
2 Answers
first, integrate the indefinite integral
with some integration by parts, you should get
the answer is
since
so the answer is
See below.
Explanation:
Filling in some of the extra steps...
You can use integration by parts.
#uv-intvdu#
Let:
#u=x^3 " " du=3x^2dx#
#dv=e^(-x)" " v=-e^(-x)#
Substituting into the above expression:
#=>-x^3e^(-x)+3intx^2e^(-x)dx#
Apply again:
#u=x^2 " " du=2xdx#
#dv=e^(-x)" " v=-e^(-x)#
#=>-x^3e^(-x)+3[-x^2e^(-x)+2intxe^(-x)dx]#
Finally:
#u=x " " du=dx#
#dv=e^(-x)" " v=-e^(-x)#
#=>-x^3e^(-x)+3[-x^2e^(-x)+2{-xe^(-x)+inte^(-x)dx}]#
Simplify:
#-e^(-x)(x^3+3x^2+6x+6)#
Evaluate from
Note that
#=>0-[-1/e^0((0)^3+3(0^2)+6(0)+6)]#
Note that
#=>1(6)=6#