integral of ((e^{2y}-y))/(e^y)

解

\int \frac{( e ^{2 y} - y )}{e ^{y}} d y

e^{y} + \frac{y}{e ^{y}} + \frac{1}{e ^{y}} + C

解答ステップ

\int \frac{e ^{2 y} - y}{e ^{y}} d y

拡張

\frac{e ^{2 y} - y}{e ^{y}} : e^{y} - \frac{y}{e ^{y}}

= \int e^{y} - \frac{y}{e ^{y}} d y

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{y} d y - \int \frac{y}{e ^{y}} d y

\int e^{y} d y = e^{y}

\int \frac{y}{e ^{y}} d y = - \frac{y}{e ^{y}} - \frac{1}{e ^{y}}

= e^{y} - (- \frac{y}{e ^{y}} - \frac{1}{e ^{y}})

簡素化

= e^{y} + \frac{y}{e ^{y}} + \frac{1}{e ^{y}}

定数を解答に追加する

= e^{y} + \frac{y}{e ^{y}} + \frac{1}{e ^{y}} + C