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find-I-n-x-0-t-n-e-xt-dt-x-gt-0-n-N-




Question Number 31095 by abdo imad last updated on 02/Mar/18
find I_n (x)= ∫_0 ^∞  t^n  e^(−xt) dt    x>0 n∈ N.
findIn(x)=0tnextdtx>0nN.
Commented by abdo imad last updated on 04/Mar/18
ch.xt=u give I_n (x)=∫_0 ^∞ ((u/x))^n  e^(−u)  (du/x)  =(1/x^(n+1) )∫_0 ^∞  u^n  e^(−u) du let put A_n =∫_0 ^∞  u^n  e^(−u) du  by parts  A_n =[−u^n e^(−u) ]_0 ^∞  +∫_0 ^∞ n u^(n−1)  e^(−u) du=nA_(n−1)  ⇒  Π_(k=1) ^n A_k =n! Π_(k=1) ^n A_(k−1)   ⇒A_n =n!A_0  =n!(  A_0 =1)⇒  I_n (x)=((n!)/x^(n+1) ) .
ch.xt=ugiveIn(x)=0(ux)neudux=1xn+10uneuduletputAn=0uneudubypartsAn=[uneu]0+0nun1eudu=nAn1k=1nAk=n!k=1nAk1An=n!A0=n!(A0=1)In(x)=n!xn+1.

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