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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Multipliziere aus

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

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

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

\int 2 d x = 2 x

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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