ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of sqrt(1-e^{2x)}

解

\int 1 - e^{2 x} d x

解

- \frac{1}{2} ln 1 - e^{2 x} + 1 + \frac{1}{2} ln 1 - e^{2 x} - 1 + 1 - e^{2 x} + C

解答ステップ

\int 1 - e^{2 x} d x

u置換積分法を適用する

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

u置換積分法を適用する

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

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

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

= - \frac{1}{2} \cdot 2 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v)

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - \frac{1}{2} \cdot 2 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

代用を戻す

= - \frac{1}{2} \cdot 2 (\frac{ln 1 - e ^{2 x} + 1}{2} - \frac{ln 1 - e ^{2 x} - 1}{2} - 1 - e^{2 x})

簡素化

- \frac{1}{2} \cdot 2 (\frac{ln 1 - e ^{2 x} + 1}{2} - \frac{ln 1 - e ^{2 x} - 1}{2} - 1 - e^{2 x}) : - \frac{1}{2} ln 1 - e^{2 x} + 1 + \frac{1}{2} ln 1 - e^{2 x} - 1 + 1 - e^{2 x}

= - \frac{1}{2} ln 1 - e^{2 x} + 1 + \frac{1}{2} ln 1 - e^{2 x} - 1 + 1 - e^{2 x}

定数を解答に追加する

= - \frac{1}{2} ln 1 - e^{2 x} + 1 + \frac{1}{2} ln 1 - e^{2 x} - 1 + 1 - e^{2 x} + C

グラフ

人気の例

limit as x approaches 0 of 3/x-2/(ln(x))

x \to 0 lim (\frac{3}{x} - \frac{2}{ln ( x )})

integral of 2pi(4x^2+4x+1)

\int 2 π (4 x^{2} + 4 x + 1) d x

limit as x approaches 2/3 of (27x^3-8)/(9x^2-4)

x \to \frac{2}{3} lim (\frac{27 x ^{3} - 8}{9 x ^{2} - 4})

derivative f(x)=(cos(x))/(x^3)

d er i v a t i v e f (x) = \frac{cos ( x )}{x ^{3}}

limit as x approaches 5 of 2x^2-7x+13

x \to 5 lim (2 x^{2} - 7 x + 13)