7.1 Integration By Parts/25

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {y}{e^{2y}}}dy\\[2ex]&u=y,-du=dy,\\[2ex]&dv={\frac {dy}{e^{2}y}}=e^{-2y}dy,v={\frac {-1}{2}}e^{-2y}\end{aligned}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }