integral from 1 to 4 of 1/(x(x-5))

Lösung

\int_{1}^{4} \frac{1}{x ( x - 5 )} d x

- \frac{4}{5} ln (2)

Dezimale

- 0.55451 \dots

Schritte zur Lösung

\int_{1}^{4} \frac{1}{x ( x - 5 )} d x

Ermittle den Partialbruch von

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

= \int_{1}^{4} - \frac{1}{5 x} + \frac{1}{5 ( x - 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

= - \int_{1}^{4} \frac{1}{5 x} d x + \int_{1}^{4} \frac{1}{5 ( x - 5 )} d x

\int_{1}^{4} \frac{1}{5 x} d x = \frac{2}{5} ln (2)

\int_{1}^{4} \frac{1}{5 ( x - 5 )} d x = - \frac{2}{5} ln (2)

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

Vereinfache

- \frac{2}{5} ln (2) - \frac{2}{5} ln (2) : - \frac{4}{5} ln (2)

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