7.1 Integration By Parts/20

$\int _{0}^{1}\left(x^{2}+1\right)e^{-x}dx$

${\begin{aligned}u&=x^{2}+1\quad dv=e^{-x}dx\\[2ex]du&=2xdx\qquad v=-e^{-x}\\[2ex]\end{aligned}}$

${\begin{aligned}&=(x^{2}+1)(-e^{-x}){\bigg |}_{0}^{1}-2\int _{0}^{1}(-e^{x})(x)dx\\[2ex]\end{aligned}}$

${\begin{aligned}u&=x\quad dv=-e^{-x}dx\\[2ex]du&=dx\qquad v=e^{-x}\\[2ex]\end{aligned}}$

${\begin{aligned}&=(x^{2}+1)(-e^{-x}){\bigg |}_{0}^{1}-2\left[(x)(e^{-x})\right]{\bigg |}_{0}^{1}-\int _{0}^{1}(e^{-x})dx\\[2ex]&=-2e^{-1}+1-{\frac {2}{e}}+2(-e^{-1}+e^{0})\\[2ex]&=-{\frac {2}{e}}+1-{\frac {2}{e}}-{\frac {2}{e}}+2\\[2ex]&=-{\frac {6}{e}}+3\end{aligned}}$