using a partial integration find a recurrence relationship between Un + 1 and Un?

Question

#Un=int_0^1x^*e^(-x)#

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Answer 1 · 2018-01-10T19:08:15

The relation is #u_(n+1)=(n+1)u_n+(1-1/e)#

#u_n=int_0^1x^(n )e^-xdx#

#u_(n+1)=int_0^1x^(n+1 )e^-xdx#

Integration by parts (Intégration par parties)

#u(x)=x^(n+1)#, #=>#, #u'(x)=(n+1)x^n#

#v'(x)=e^-x#, #=>#, #v(x)=-e^-x#

#intuv'=uv-intu'v#

#u_(n+1)=int_0^1x^(n+1 )e^-xdx#

#=[-e^-x x^(n+1)]_0^1+(n+1)intx^n e^-xdx#

#=(-1/e+1)+(n+1)u_n#

#u_(n+1)=(n+1)u_n+(1-1/e)#

J'espère que je vous ai aidé.