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Beliebt Rechnen >

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

Lösung

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

Lösung

- ln (5) + 3 ln (2)

+1

Dezimale

0.47000 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

= 2 \cdot \int_{1}^{2} \frac{1}{x} - \frac{x}{x ^{2} + 1} 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

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

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

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

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

Vereinfache

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

= - ln (5) + 3 ln (2)

Graph

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

derivative of 2x^3-15x^2+36x+5

\frac{d}{d x} (2 x^{3} - 15 x^{2} + 36 x + 5)