# Using integration by parts, we find that int x^(n)e^(-x) dx=

Tushar Chandra | Certified Educator

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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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