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人気のある微分積分 >

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

解

\int \frac{e ^{2 x}}{1 + e ^{x}} d x

解

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

解答ステップ

\int \frac{e ^{2 x}}{1 + e ^{x}} d x

u置換積分法を適用する

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

拡張

\frac{u - 1}{u} : u - \frac{1}{u}

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{2}{3} u^{\frac{3}{2}} - 2 u

代用を戻す

u = 1 + e^{x}

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

定数を解答に追加する

= \frac{2}{3} (1 + e^{x})^{\frac{3}{2}} - 2 1 + e^{x} + C

グラフ

人気の例

integral from 0 to 2pi of cos(x)sin(x)

\int_{0}^{2 π} cos (x) sin (x) d x

limit as x approaches-6 of 11x+3

x \to - 6 lim (11 x + 3)

tangent f(x)=12-x^2,(3,3)

t an g e n t f (x) = 12 - x^{2}, (3, 3)

tangent f(x)=(18)/(x^2+2),(1,6)

t an g e n t f (x) = \frac{18}{x ^{2} + 2}, (1, 6)

limit as x approaches 0+of x^{11x}

x \to 0 + lim (x^{11 x})