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

解答

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

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

求解步骤

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

乘开

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

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

使用积分加法定则:

\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}})

化简

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

解答补常数

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

d er i v a t i v e (x^{2} - 12) (2 x - 9)

\int_{x}^{2} 0.5 d y

\int (sin (x))^{5} d x

\int x^{2} \cdot e^{5 x} d x

n = - 1 \sum \infty (\frac{1}{2})^{n}