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人気のある微分積分 >

integral of (2+e^x)/(e^x)

解

\int \frac{2 + e ^{x}}{e ^{x}} d x

解

- \frac{2}{e ^{x}} + x + C

解答ステップ

\int \frac{2 + e ^{x}}{e ^{x}} d x

u置換積分法を適用する

= \int \frac{2 + u}{u ^{2}} d u

拡張

\frac{2 + u}{u ^{2}} : \frac{2}{u ^{2}} + \frac{1}{u}

= \int \frac{2}{u ^{2}} + \frac{1}{u} d u

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 \frac{2}{u ^{2}} d u + \int \frac{1}{u} d u

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

\int \frac{1}{u} d u = ln ∣ u ∣

= - \frac{2}{u} + ln ∣ u ∣

代用を戻す

u = e^{x}

= - \frac{2}{e ^{x}} + ln ∣ e^{x} ∣

ln ∣ e^{x} ∣ = x

= - \frac{2}{e ^{x}} + x

定数を解答に追加する

= - \frac{2}{e ^{x}} + x + C

グラフ

人気の例

sum from n=1 to infinity of 7/(2^{n-1)}

n = 1 \sum \infty \frac{7}{2 ^{n - 1}}

derivative of 2e^{2x}+3e^{3x}

\frac{d}{d x} (2 e^{2 x} + 3 e^{3 x})

limit as x approaches a of [f(x)]^{g(x)}

x \to a lim ([f (x)]^{g (x)})

limit as t approaches 0 of 3/t-3/(e^t-1)

t \to 0 lim (\frac{3}{t} - \frac{3}{e ^{t} - 1})

y^'+y=te^{-t}+1

y^{^{'}} + y = t e^{- t} + 1