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 (e^{2x})/(e^{2x)-e^x-6}

解

\int \frac{e ^{2 x}}{e ^{2 x} - e ^{x} - 6} d x

解

2 (\frac{1}{4} ln - 4 e^{x} + 4 e^{2 x} - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} e^{x} - ln - \frac{6}{5} + \frac{2}{5} e^{x})) + C

解答ステップ

\int \frac{e ^{2 x}}{e ^{2 x} - e ^{x} - 6} d x

u置換積分法を適用する

= \int \frac{u}{u ^{2} - u - 6} d u

平方完成する

u^{2} - u - 6 : (u - \frac{1}{2})^{2} - \frac{25}{4}

= \int \frac{u}{( u - \frac{1}{2} ) ^{2} - \frac{25}{4}} d u

u置換積分法を適用する

= \int \frac{2 ( 2 v + 1 )}{4 v ^{2} - 25} d v

定数を除く:

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

= 2 \cdot \int \frac{2 v + 1}{4 v ^{2} - 25} d v

拡張

\frac{2 v + 1}{4 v ^{2} - 25} : \frac{2 v}{4 v ^{2} - 25} + \frac{1}{4 v ^{2} - 25}

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

= 2 (\int \frac{2 v}{4 v ^{2} - 25} d v + \int \frac{1}{4 v ^{2} - 25} d v)

\int \frac{2 v}{4 v ^{2} - 25} d v = \frac{1}{4} ln 4 v^{2} - 25

\int \frac{1}{4 v ^{2} - 25} d v = - \frac{1}{20} (ln \frac{2}{5} v + 1 - ln \frac{2}{5} v - 1)

= 2 (\frac{1}{4} ln 4 v^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} v + 1 - ln \frac{2}{5} v - 1))

代用を戻す

= 2 (\frac{1}{4} ln 4 (e^{x} - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (e^{x} - \frac{1}{2}) + 1 - ln \frac{2}{5} (e^{x} - \frac{1}{2}) - 1))

簡素化

2 (\frac{1}{4} ln 4 (e^{x} - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (e^{x} - \frac{1}{2}) + 1 - ln \frac{2}{5} (e^{x} - \frac{1}{2}) - 1)) : 2 (\frac{1}{4} ln - 4 e^{x} + 4 e^{2 x} - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} e^{x} - ln - \frac{6}{5} + \frac{2}{5} e^{x}))

= 2 (\frac{1}{4} ln - 4 e^{x} + 4 e^{2 x} - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} e^{x} - ln - \frac{6}{5} + \frac{2}{5} e^{x}))

定数を解答に追加する

= 2 (\frac{1}{4} ln - 4 e^{x} + 4 e^{2 x} - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} e^{x} - ln - \frac{6}{5} + \frac{2}{5} e^{x})) + C

グラフ

人気の例

integral of (cos^3(8x))/(sin^4(8x))

\int \frac{cos ^{3} ( 8 x )}{sin ^{4} ( 8 x )} d x

integral of (e^{aln(x)}+e^{xln(a)})

\int (e^{a l n (x)} + e^{x l n (a)}) d x

integral of e^x(4-e^x)^{2/3}

\int e^{x} (4 - e^{x})^{\frac{2}{3}} d x

integral of ((y^3+4y))/(9+y^4)

\int \frac{( y ^{3} + 4 y )}{9 + y ^{4}} d y

integral of (cos(θ))/(sqrt(4-sin^2(θ)))

\int \frac{cos ( θ )}{4 - sin ^{2} ( θ )} d θ