de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of ((x-1)^3)/((x+2)^5)

Lösung

\int \frac{( x - 1 ) ^{3}}{( x + 2 ) ^{5}} d x

Lösung

- \frac{1}{x + 2} + \frac{9}{2 ( x + 2 ) ^{2}} - \frac{9}{( x + 2 ) ^{3}} - 27 (\frac{x}{( x + 2 ) ^{5}} - \frac{5}{4 ( x + 2 ) ^{4}} + \frac{2}{( x + 2 ) ^{5}}) + C

Schritte zur Lösung

\int \frac{( x - 1 ) ^{3}}{( x + 2 ) ^{5}} d x

Ermittle den Partialbruch von

\frac{( x - 1 ) ^{3}}{( x + 2 ) ^{5}} : \frac{1}{( x + 2 ) ^{2}} - \frac{9}{( x + 2 ) ^{3}} + \frac{27}{( x + 2 ) ^{4}} - \frac{27}{( x + 2 ) ^{5}}

= \int \frac{1}{( x + 2 ) ^{2}} - \frac{9}{( x + 2 ) ^{3}} + \frac{27}{( x + 2 ) ^{4}} - \frac{27}{( x + 2 ) ^{5}} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{( x + 2 ) ^{2}} d x - \int \frac{9}{( x + 2 ) ^{3}} d x + \int \frac{27}{( x + 2 ) ^{4}} d x - \int \frac{27}{( x + 2 ) ^{5}} d x

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

\int \frac{9}{( x + 2 ) ^{3}} d x = - \frac{9}{2 ( x + 2 ) ^{2}}

\int \frac{27}{( x + 2 ) ^{4}} d x = - \frac{9}{( x + 2 ) ^{3}}

\int \frac{27}{( x + 2 ) ^{5}} d x = 27 (\frac{x}{( x + 2 ) ^{5}} - \frac{5}{4 ( x + 2 ) ^{4}} + \frac{2}{( x + 2 ) ^{5}})

= - \frac{1}{x + 2} - (- \frac{9}{2 ( x + 2 ) ^{2}}) - \frac{9}{( x + 2 ) ^{3}} - 27 (\frac{x}{( x + 2 ) ^{5}} - \frac{5}{4 ( x + 2 ) ^{4}} + \frac{2}{( x + 2 ) ^{5}})

Vereinfache

= - \frac{1}{x + 2} + \frac{9}{2 ( x + 2 ) ^{2}} - \frac{9}{( x + 2 ) ^{3}} - 27 (\frac{x}{( x + 2 ) ^{5}} - \frac{5}{4 ( x + 2 ) ^{4}} + \frac{2}{( x + 2 ) ^{5}})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{x + 2} + \frac{9}{2 ( x + 2 ) ^{2}} - \frac{9}{( x + 2 ) ^{3}} - 27 (\frac{x}{( x + 2 ) ^{5}} - \frac{5}{4 ( x + 2 ) ^{4}} + \frac{2}{( x + 2 ) ^{5}}) + C

Graph

Beliebte Beispiele

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

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

integral 1/(4-x)

in t e g r a l \frac{1}{4 - x}

integral of (x^2+4x-5)

\int (x^{2} + 4 x - 5) d x

integral of cos(14x)sin(8x)

\int cos (14 x) sin (8 x) d x

integral of sec^3(x/2)

\int sec^{3} (\frac{x}{2}) d x