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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Vereinfache

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

Graph

Beliebte Beispiele

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

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

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

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\int \frac{1}{( w ^{2} + 4 ) ^{\frac{3}{2}}} d w

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\int 4 x^{2} sin (x^{3}) d x