integral from 1 to 2 of 9/(x^3+5x)

Lösung

\int_{1}^{2} \frac{9}{x ^{3} + 5 x} d x

9 (\frac{1}{5} ln (2) - \frac{1}{10} ln (\frac{3}{2}))

Dezimale

0.88274 \dots

Schritte zur Lösung

\int_{1}^{2} \frac{9}{x ^{3} + 5 x} d x

Entferne die Konstante:

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

= 9 \cdot \int_{1}^{2} \frac{1}{x ^{3} + 5 x} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} + 5 x} : \frac{1}{5 x} - \frac{x}{5 ( x ^{2} + 5 )}

= 9 \cdot \int_{1}^{2} \frac{1}{5 x} - \frac{x}{5 ( x ^{2} + 5 )} 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

= 9 (\int_{1}^{2} \frac{1}{5 x} d x - \int_{1}^{2} \frac{x}{5 ( x ^{2} + 5 )} d x)

\int_{1}^{2} \frac{1}{5 x} d x = \frac{1}{5} ln (2)

\int_{1}^{2} \frac{x}{5 ( x ^{2} + 5 )} d x = \frac{1}{10} ln (\frac{3}{2})

= 9 (\frac{1}{5} ln (2) - \frac{1}{10} ln (\frac{3}{2}))

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