integral of (x+2)/((x^2+4x+6)^5)

Lösung

\int \frac{x + 2}{( x ^{2} + 4 x + 6 ) ^{5}} d x

- \frac{1}{8 ( x ^{2} + 4 x + 6 ) ^{4}} + C

Schritte zur Lösung

\int \frac{x + 2}{( x ^{2} + 4 x + 6 ) ^{5}} d x

Wende U-Substitution an

= \int \frac{u}{( u ^{2} + 2 ) ^{5}} d u

Wende U-Substitution an

= \int \frac{1}{2 ( v + 2 ) ^{5}} d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{( v + 2 ) ^{5}} d v

Wende die partielle Integration an

= \frac{1}{2} (\frac{v}{( v + 2 ) ^{5}} - \int - \frac{5 v}{( v + 2 ) ^{6}} d v)

\int - \frac{5 v}{( v + 2 ) ^{6}} d v = \frac{5}{4 ( v + 2 ) ^{4}} - \frac{2}{( v + 2 ) ^{5}}

= \frac{1}{2} (\frac{v}{( v + 2 ) ^{5}} - (\frac{5}{4 ( v + 2 ) ^{4}} - \frac{2}{( v + 2 ) ^{5}}))

Ersetze zurück

= \frac{1}{2} \frac{( x + 2 ) ^{2}}{( ( x + 2 ) ^{2} + 2 ) ^{5}} - \frac{5}{4 ( ( x + 2 ) ^{2} + 2 ) ^{4}} - \frac{2}{( ( x + 2 ) ^{2} + 2 ) ^{5}}

Vereinfache

\frac{1}{2} \frac{( x + 2 ) ^{2}}{( ( x + 2 ) ^{2} + 2 ) ^{5}} - \frac{5}{4 ( ( x + 2 ) ^{2} + 2 ) ^{4}} - \frac{2}{( ( x + 2 ) ^{2} + 2 ) ^{5}} : - \frac{1}{8 ( x ^{2} + 4 x + 6 ) ^{4}}

= - \frac{1}{8 ( x ^{2} + 4 x + 6 ) ^{4}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{8 ( x ^{2} + 4 x + 6 ) ^{4}} + C

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