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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

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

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

= - 2 ln ∣ u ∣ - \frac{1}{u} + 2 ln ∣ u + 1 ∣ - \frac{1}{u + 1}

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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