integral of (x^4+9)/(x^2+9x^2)

Lösung

\int \frac{x ^{4} + 9}{x ^{2} + 9 x ^{2}} d x

\frac{1}{10} (\frac{x ^{3}}{3} - \frac{9}{x}) + C

Schritte zur Lösung

\int \frac{x ^{4} + 9}{x ^{2} + 9 x ^{2}} d x

Vereinfache

= \int \frac{x ^{4} + 9}{10 x ^{2}} d x

Entferne die Konstante:

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

= \frac{1}{10} \cdot \int \frac{x ^{4} + 9}{x ^{2}} d x

Multipliziere aus

\frac{x ^{4} + 9}{x ^{2}} : x^{2} + \frac{9}{x ^{2}}

= \frac{1}{10} \cdot \int x^{2} + \frac{9}{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

= \frac{1}{10} (\int x^{2} d x + \int \frac{9}{x ^{2}} d x)

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

\int \frac{9}{x ^{2}} d x = - \frac{9}{x}

= \frac{1}{10} (\frac{x ^{3}}{3} - \frac{9}{x})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} (\frac{x ^{3}}{3} - \frac{9}{x}) + C

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