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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

\int \frac{3}{x + 2} d x = 3 ln ∣ x + 2 ∣

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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