∫ 0 1 y e 2 y d y {\displaystyle \int _{0}^{1}{\frac {y}{e^{2y}}}dy}
u = y d u = d y {\displaystyle u=y\qquad du=dy}
d v = d y e 2 y = e − 2 y d y v = − 1 2 e − 2 y {\displaystyle dv={\frac {dy}{e^{2}y}}=e^{-2y}dy\qquad v={\frac {-1}{2}}e^{-2y}}
∫ 0 1 y e 2 y d y = [ − 1 2 y e − 2 y ] | 0 1 + 1 2 ∫ 0 1 e − 2 y d y = ( − 1 2 e − 2 + 0 ) − 1 4 [ e − 2 y ] | 0 1 = − 1 2 e − 2 − 1 4 e − 2 + 1 4 = 1 4 − 3 4 e − 2 {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {y}{e^{2y}}}dy&=\left[{\frac {-1}{2}}ye^{-2y}\right]{\bigg |}_{0}^{1}+{\frac {1}{2}}\int _{0}{1}e^{-2y}dy\\[2ex]&=\left({\frac {-1}{2}}e^{-2}+0\right)-{\frac {1}{4}}\left[e^{-2y}\right]{\bigg |}_{0}^{1}\\[2ex]&={\frac {-1}{2}}e^{-2}-{\frac {1}{4}}e^{-2}+{\frac {1}{4}}\\[2ex]&={\frac {1}{4}}-{\frac {3}{4}}e^{-2}\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {y}{e^{2y}}}dy&=\left[{\frac {-1}{2}}ye^{-2y}\right]{\bigg |}_{0}^{1}+{\frac {1}{2}}\int _{0}{1}e^{-2y}dy\\[2ex]&=\left({\frac {-1}{2}}e^{-2}+0\right)-{\frac {1}{4}}\left[e^{-2y}\right]{\bigg |}_{0}^{1}\\[2ex]&={\frac {-1}{2}}e^{-2}-{\frac {1}{4}}e^{-2}+{\frac {1}{4}}\\[2ex]&={\frac {1}{4}}-{\frac {3}{4}}e^{-2}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fe2e33184a069db1848fc7fdf750f0d1bcdf0d4e)