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

Lösung

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

5 \cdot \frac{4 ln ( 2 ) - ln ( 7 )}{6}

Dezimale

0.68889 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

= 5 \cdot \int_{1}^{2} \frac{1}{3 x} - \frac{x}{3 ( x ^{2} + 3 )} 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

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

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

\int_{1}^{2} \frac{x}{3 ( x ^{2} + 3 )} d x = \frac{1}{6} (ln (7) - 2 ln (2))

= 5 (\frac{1}{3} ln (2) - \frac{1}{6} (ln (7) - 2 ln (2)))

Füge

\frac{1}{3} ln (2) - \frac{1}{6} (ln (7) - 2 ln (2))

zusammen:

\frac{4 ln ( 2 ) - ln ( 7 )}{6}

= 5 \cdot \frac{4 ln ( 2 ) - ln ( 7 )}{6}

\int ((3 x^{5} - 4)^{2} \cdot 5 x^{4}) d x

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