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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{2}{y - 2} + ln ∣ y ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{y ^{2} - 2 y + 4}{y ( y - 2 ) ^{2}} : \frac{y ^{2}}{y ( y - 2 ) ^{2}} - \frac{2 y}{y ( y - 2 ) ^{2}} + \frac{4}{y ( y - 2 ) ^{2}}

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

\int \frac{y ^{2}}{y ( y - 2 ) ^{2}} d y = ln ∣ y - 2 ∣ - \frac{2}{y - 2}

\int \frac{2 y}{y ( y - 2 ) ^{2}} d y = - \frac{2}{y - 2}

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

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

Vereinfache

ln ∣ y - 2 ∣ - \frac{2}{y - 2} - (- \frac{2}{y - 2}) + ln ∣ y ∣ - ln ∣ y - 2 ∣ - \frac{2}{y - 2} : - \frac{2}{y - 2} + ln ∣ y ∣

= - \frac{2}{y - 2} + ln ∣ y ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{y - 2} + ln ∣ y ∣ + C

Graph

Beliebte Beispiele

y^{''''}-y^{''}=4x+2xe^{-x}

y^{^{''''}} - y^{^{''}} = 4 x + 2 x e^{- x}

steigung (1.4)(7.1)

s l o p e (1.4) (7.1)

integral from 3 to 9 of 1/(x(ln(x))^2)

\int_{3}^{9} \frac{1}{x ( ln ( x ) ) ^{2}} d x

derivative f(x)=(1-4x)/(3+x)

d er i v a t i v e f (x) = \frac{1 - 4 x}{3 + x}

limit as x approaches 5 of ln(x-4)+1/2

x \to 5 lim (ln (x - 4) + \frac{1}{2})