de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of ((y^4-8))/(y^3+2y^2)

Lösung

\int \frac{( y ^{4} - 8 )}{y ^{3} + 2 y ^{2}} d y

Lösung

\frac{y ^{2}}{2} - 2 y + 2 ln ∣ y ∣ + \frac{4}{y} + 2 ln ∣ y + 2 ∣ + C

Schritte zur Lösung

\int \frac{y ^{4} - 8}{y ^{3} + 2 y ^{2}} d y

Ermittle den Partialbruch von

\frac{y ^{4} - 8}{y ^{3} + 2 y ^{2}} : y - 2 + \frac{2}{y} - \frac{4}{y ^{2}} + \frac{2}{y + 2}

= \int y - 2 + \frac{2}{y} - \frac{4}{y ^{2}} + \frac{2}{y + 2} d y

Wende die Summenregel an:

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

= \int y d y - \int 2 d y + \int \frac{2}{y} d y - \int \frac{4}{y ^{2}} d y + \int \frac{2}{y + 2} d y

\int y d y = \frac{y ^{2}}{2}

\int 2 d y = 2 y

\int \frac{2}{y} d y = 2 ln ∣ y ∣

\int \frac{4}{y ^{2}} d y = - \frac{4}{y}

\int \frac{2}{y + 2} d y = 2 ln ∣ y + 2 ∣

= \frac{y ^{2}}{2} - 2 y + 2 ln ∣ y ∣ - (- \frac{4}{y}) + 2 ln ∣ y + 2 ∣

Vereinfache

= \frac{y ^{2}}{2} - 2 y + 2 ln ∣ y ∣ + \frac{4}{y} + 2 ln ∣ y + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{y ^{2}}{2} - 2 y + 2 ln ∣ y ∣ + \frac{4}{y} + 2 ln ∣ y + 2 ∣ + C

Graph

Beliebte Beispiele

integral of csc(1-2x)cot(1-2x)

\int csc (1 - 2 x) cot (1 - 2 x) d x

integral of (7x-1)^6

\int (7 x - 1)^{6} d x

integral of 5/(sqrt(4-9x^2))

\int \frac{5}{4 - 9 x ^{2}} d x

integral of 2/(sqrt(5-4x-x^2))

\int \frac{2}{5 - 4 x - x ^{2}} d x

integral of 1/(2cos^2(3x))

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