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Beliebt Rechnen >

integral of (1+1/x)^{-3}*1/(x^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende die partielle Integration an

= - \frac{x ^{2}}{( x + 1 ) ^{3}} - \int - \frac{3 x}{( x + 1 ) ^{4}} d x

\int - \frac{3 x}{( x + 1 ) ^{4}} d x = \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{( x + 1 ) ^{3}}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

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\int \frac{1}{( ln ( x ) )} d x