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 (11w+4)/((3+6w-w^2)^2)

解

\int \frac{11 w + 4}{( 3 + 6 w - w ^{2} ) ^{2}} d w

解

\frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )}) + C

解答ステップ

\int \frac{11 w + 4}{( 3 + 6 w - w ^{2} ) ^{2}} d w

平方完成する

3 + 6 w - w^{2} : - (w - 3)^{2} + 12

= \int \frac{11 w + 4}{( - ( w - 3 ) ^{2} + 12 ) ^{2}} d w

u置換積分法を適用する

= \int \frac{11 u + 37}{( - u ^{2} + 12 ) ^{2}} d u

拡張

\frac{11 u + 37}{( - u ^{2} + 12 ) ^{2}} : \frac{11 u}{( - u ^{2} + 12 ) ^{2}} + \frac{37}{( - u ^{2} + 12 ) ^{2}}

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{11 u}{( - u ^{2} + 12 ) ^{2}} d u + \int \frac{37}{( - u ^{2} + 12 ) ^{2}} d u

\int \frac{11 u}{( - u ^{2} + 12 ) ^{2}} d u = \frac{11}{2 ( - u ^{2} + 12 )}

\int \frac{37}{( - u ^{2} + 12 ) ^{2}} d u = 37 (\frac{1}{96 3} ln u + 23 - \frac{1}{48 ( u + 2 3 )} - \frac{1}{96 3} ln u - 23 - \frac{1}{48 ( u - 2 3 )})

= \frac{11}{2 ( - u ^{2} + 12 )} + 37 (\frac{1}{96 3} ln u + 23 - \frac{1}{48 ( u + 2 3 )} - \frac{1}{96 3} ln u - 23 - \frac{1}{48 ( u - 2 3 )})

代用を戻す

u = w - 3

= \frac{11}{2 ( - ( w - 3 ) ^{2} + 12 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )})

拡張

- (w - 3)^{2} + 12 : - w^{2} + 6 w + 3

= \frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )})

定数を解答に追加する

= \frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )}) + C

グラフ

人気の例

integral of 1/((2-3x)^4)

\int \frac{1}{( 2 - 3 x ) ^{4}} d x

integral of (x^2-1)/(x^2-16)

\int \frac{x ^{2} - 1}{x ^{2} - 16} d x

integral of cos^6(4x)

\int cos^{6} (4 x) d x

integral of (sqrt(12+x^2))/x

\int \frac{12 + x ^{2}}{x} d x

integral of (-2x^{1/2})

\int (- 2 x^{\frac{1}{2}}) d x