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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{3} (- \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2})) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{3 ( x + 1 )} d x = \frac{1}{3} ln ∣ x + 1 ∣

\int \frac{- x + 1}{3 ( x ^{2} + 2 )} d x = \frac{1}{3} (- \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2}))

= \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{3} (- \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{3} (- \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2})) + C

Graph

Beliebte Beispiele

integral of (sqrt(u-1))/u

\int \frac{u - 1}{u} d u

integral of sqrt(e^{2x)+e^{2x}}

\int e^{2 x} + e^{2 x} d x

integral of (cos(x))/(sqrt(3-sin^2(x)))

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

integral of (x+9)/(x(x+2))

\int \frac{x + 9}{x ( x + 2 )} d x

integral of (u^{3/2}-3u+14)

\int (u^{\frac{3}{2}} - 3 u + 14) d u