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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{2} ln ∣ x ∣ + \frac{9}{10} ln ∣ 2 x - 1 ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2} + 2 x + 1}{2 x ^{3} + 3 x ^{2} - 2 x} : - \frac{1}{2 x} + \frac{9}{5 ( 2 x - 1 )} + \frac{1}{10 ( x + 2 )}

= \int - \frac{1}{2 x} + \frac{9}{5 ( 2 x - 1 )} + \frac{1}{10 ( x + 2 )} 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}{2 x} d x + \int \frac{9}{5 ( 2 x - 1 )} d x + \int \frac{1}{10 ( x + 2 )} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{9}{5 ( 2 x - 1 )} d x = \frac{9}{10} ln ∣ 2 x - 1 ∣

\int \frac{1}{10 ( x + 2 )} d x = \frac{1}{10} ln ∣ x + 2 ∣

= - \frac{1}{2} ln ∣ x ∣ + \frac{9}{10} ln ∣ 2 x - 1 ∣ + \frac{1}{10} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ x ∣ + \frac{9}{10} ln ∣ 2 x - 1 ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + C

Graph

Beliebte Beispiele

integral of (3sin(2x))/(cos(2x)+2)

\int \frac{3 sin ( 2 x )}{cos ( 2 x ) + 2} d x

integral of (2x-3)/(sqrt(x))

\int \frac{2 x - 3}{x} d x

integral of cos(ax)e^{-sx}

\int cos (a x) e^{- s x} d x

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

\int \frac{4 - x}{x ^{3}} d x

integral of (12x+20)/(x^2+2x+1)

\int \frac{12 x + 20}{x ^{2} + 2 x + 1} d x