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

Lösung

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

ln (5) - ln (2)

Dezimale

0.91629 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Vereinfache

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

= ln (5) - ln (2)