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Beliebt Rechnen >

integral of (6x+5)/(6x^2)

Lösung

\int \frac{6 x + 5}{6 x ^{2}} d x

Lösung

ln ∣ x ∣ - \frac{5}{6 x} + C

Schritte zur Lösung

\int \frac{6 x + 5}{6 x ^{2}} d x

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{6 x + 5}{x ^{2}} d x

Multipliziere aus

\frac{6 x + 5}{x ^{2}} : \frac{6}{x} + \frac{5}{x ^{2}}

= \frac{1}{6} \cdot \int \frac{6}{x} + \frac{5}{x ^{2}} 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

= \frac{1}{6} (\int \frac{6}{x} d x + \int \frac{5}{x ^{2}} d x)

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{5}{x ^{2}} d x = - \frac{5}{x}

= \frac{1}{6} (6 ln ∣ x ∣ - \frac{5}{x})

Vereinfache

\frac{1}{6} (6 ln ∣ x ∣ - \frac{5}{x}) : ln ∣ x ∣ - \frac{5}{6 x}

= ln ∣ x ∣ - \frac{5}{6 x}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - \frac{5}{6 x} + C

Graph

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