integral of x/(3x-1)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u + 1}{9 u} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int \frac{u + 1}{u} d u

Multipliziere aus

\frac{u + 1}{u} : 1 + \frac{1}{u}

= \frac{1}{9} \cdot \int 1 + \frac{1}{u} d u

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}{9} (\int 1 d u + \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{9} (u + ln ∣ u ∣)

Setze in

u = 3 x - 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

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

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

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