Using integration by parts, we find that `int x^(n)e^(-x) dx=`
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The integral `int x^n e^-x dx` has to be determined.
Integration by parts gives us the rule: `int u dv = u*v - int v du`
Let `u = x^n` and `dv = e^-x dx`
`du = n*x^(n-1) dx`
`v = -e^(-x)`
`int x^n e^-x dx`
`= x^n*-1*e^-x - int -1*e^-x*n*x^(n-1) dx`
= `x^n*-1*e^-x + n*int e^-x*x^(n-1) dx`
= `-x^n*e^-x + n*int e^-x*x^(n-1) dx`
`int e^-x*x^(n-1) dx`
= `-x^(n-1)*e^-x + (n-1)*int e^-x*x^(n-2) dx`
Substituting this in the original integral
`int x^n*e^-x dx`
= `-x^n*e^-x + n*(-x^(n-1)*e^-x + (n-1)*int e^-x*x^(n-2) dx)`
= `-x^n*e^-x- n*x^(n-1)*e^-x + n*(n-1)*int e^-x*x^(n-2) dx`
= `-e^-x*(x^n+n*x^(n-1))+n*(n-1)*int e^-x*x^(n-2) dx`
This can be continued n times to yield the final result.
= `-e^-x*(x^n+n*x^(n-1)+ n*(n-1)x^(n-2)+...n!)`
The integral...
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