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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

\frac{u - 2}{u ^{3}} : \frac{1}{u ^{2}} - \frac{2}{u ^{3}}

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

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

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

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

Setze in

u = 2 + x

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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