∫ 0 1 ( x 2 + 1 ) e − x d x {\displaystyle \int _{0}^{1}\left(x^{2}+1\right)e^{-x}dx}
u = x 2 + 1 d v = e − x d x d u = 2 x d x v = − e − x {\displaystyle {\begin{aligned}u&=x^{2}+1\quad dv=e^{-x}dx\\[2ex]du&=2xdx\qquad v=-e^{-x}\\[2ex]\end{aligned}}}
= ( x 2 + 1 ) ( − e − x ) | 0 1 − 2 ∫ 0 1 ( − e x ) ( x ) d x {\displaystyle {\begin{aligned}&=(x^{2}+1)(-e^{-x}){\bigg |}_{0}^{1}-2\int _{0}^{1}(-e^{x})(x)dx\\[2ex]\end{aligned}}}
u = x d v = − e − x d x d u = d x v = e − x {\displaystyle {\begin{aligned}u&=x\quad dv=-e^{-x}dx\\[2ex]du&=dx\qquad v=e^{-x}\\[2ex]\end{aligned}}}
= ( x 2 + 1 ) ( − e − x ) | 0 1 − 2 [ ( x ) ( e − x ) ] | 0 1 − ∫ 0 1 ( e − x ) d x = − 2 e − 1 + 1 − 2 e + 2 ( − e − 1 + e 0 ) = − 2 e + 1 − 2 e − 2 e + 2 {\displaystyle {\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})&=-{\frac {2}{e}}+1-{\frac {2}{e}}-{\frac {2}{e}}+2\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=x^{2}+1\quad dv=e^{-x}dx\\[2ex]du&=2xdx\qquad v=-e^{-x}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dca2a9cf6138c41cb61d56c1ee91c9c4fea73909)
![{\displaystyle {\begin{aligned}&=(x^{2}+1)(-e^{-x}){\bigg |}_{0}^{1}-2\int _{0}^{1}(-e^{x})(x)dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/06474884d0ba6d6fc8fae829d72953453d786c3a)
![{\displaystyle {\begin{aligned}u&=x\quad dv=-e^{-x}dx\\[2ex]du&=dx\qquad v=e^{-x}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cfe5f1427cf8d21f06f446c8bf4b0a6dc88f5acf)
![{\displaystyle {\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})&=-{\frac {2}{e}}+1-{\frac {2}{e}}-{\frac {2}{e}}+2\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8ac79ef39f9751580cad41aaf7fbf66c108f33e5)