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Question Number 31095 by abdo imad last updated on 02/Mar/18
findIn(x)=∫0∞tne−xtdtx>0n∈N.
Commented byabdo imad last updated on 04/Mar/18
ch.xt=ugiveIn(x)=∫0∞(ux)ne−udux =1xn+1∫0∞une−uduletputAn=∫0∞une−udubyparts An=[−une−u]0∞+∫0∞nun−1e−udu=nAn−1⇒ ∏k=1nAk=n!∏k=1nAk−1⇒An=n!A0=n!(A0=1)⇒ In(x)=n!xn+1.
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