integral of x/(3x+5)

Lösung

\int \frac{x}{3 x + 5} d x

\frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣ + C

Schritte zur Lösung

\int \frac{x}{3 x + 5} d x

Lange Division

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

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

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{5}{3 ( 3 x + 5 )} d x = \frac{5}{9} ln ∣ 3 x + 5 ∣

= \frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣ + C

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