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人気のある微分積分 >

integral of (e^x)/(e^{3x)+e^{2x}}

解

\int \frac{e ^{x}}{e ^{3 x} + e ^{2 x}} d x

解

- x - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣ + C

解答ステップ

\int \frac{e ^{x}}{e ^{3 x} + e ^{2 x}} d x

簡素化

\frac{e ^{x}}{e ^{3 x} + e ^{2 x}} : \frac{1}{e ^{x} ( e ^{x} + 1 )}

= \int \frac{1}{e ^{x} ( e ^{x} + 1 )} d x

u置換積分法を適用する

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

以下の部分分数を得る:

\frac{1}{u ^{2} ( u + 1 )} : - \frac{1}{u} + \frac{1}{u ^{2}} + \frac{1}{u + 1}

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= - ln ∣ u ∣ - \frac{1}{u} + ln ∣ u + 1 ∣

代用を戻す

u = e^{x}

= - ln ∣ e^{x} ∣ - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣

ln ∣ e^{x} ∣ = x

= - x - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣

定数を解答に追加する

= - x - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣ + C

グラフ

人気の例

(\partial)/(\partial y)(|y|+1/(2a)|x-y|^2)

\frac{\partial}{\partial y} (∣ y ∣ + \frac{1}{2 a} ∣ x - y ∣^{2})

integral of 1/(4x+1)

\int \frac{1}{4 x + 1} d x

(\partial)/(\partial x)(7/x)

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limit as x approaches 4+of (x+1)/(x-4)

x \to 4 + lim (\frac{x + 1}{x - 4})

(\partial)/(\partial y)(3x^2y^2+2)

\frac{\partial}{\partial y} (3 x^{2} y^{2} + 2)