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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

(1 + \frac{1}{x})^{2} \frac{1}{x ^{2}} : \frac{( x + 1 ) ^{2}}{x ^{4}}

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

Multipliziere aus

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

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

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}{x ^{2}} d x + \int \frac{2}{x ^{3}} d x + \int \frac{1}{x ^{4}} d x

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

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

= - \frac{1}{x} - \frac{1}{x ^{2}} - \frac{1}{3 x ^{3}}

Füge eine Konstante zur Lösung hinzu

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

Graph

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