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Beliebt Rechnen >

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

Lösung

integral

\frac{x ^{2}}{( x + 2 ) ^{3}}

Lösung

ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 2 ) ^{2}}{u ^{3}} : \frac{1}{u} - \frac{4}{u ^{2}} + \frac{4}{u ^{3}}

= \int \frac{1}{u} - \frac{4}{u ^{2}} + \frac{4}{u ^{3}} 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{1}{u} d u - \int \frac{4}{u ^{2}} d u + \int \frac{4}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{4}{u ^{2}} d u = - \frac{4}{u}

\int \frac{4}{u ^{3}} d u = - \frac{2}{u ^{2}}

= ln ∣ u ∣ - (- \frac{4}{u}) - \frac{2}{u ^{2}}

Setze in

u = x + 2

ein

= ln ∣ x + 2 ∣ - (- \frac{4}{x + 2}) - \frac{2}{( x + 2 ) ^{2}}

Vereinfache

= ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}} + C

Graph

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