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MAT$137$Y$1$ Y LEC$51012025...$

MAT$137$ Pset $4$

Due: July ${18}^{th}2025$

page $12$ of $12$

$8.($a$)$ Prove that for any $n0,$ we have

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^n{e}^{x}dx=x^n{e}^x-n{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^{n-1}{e}^{x}dx$

$($b$)$ Prove that if $p$:$RR$ is a degree $n$ polynomial, then

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}p(x)e^{x}dx=e^x{\displaystyle \underset{k=0}{\overset{n}{}}}(-1{)}^k{p}^{(k)}(x)C$

$=e^x[p(x)-p^{\text{'}}(x)$ cdots $(-1{)}^n{p}^{(n)}(x)]C,$

for some CinR.

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