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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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