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Beliebt Rechnen >

integral of 6/(2x+3x^2)

Lösung

\int \frac{6}{2 x + 3 x ^{2}} d x

Lösung

6 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ 3 x + 2 ∣) + C

Schritte zur Lösung

\int \frac{6}{2 x + 3 x ^{2}} d x

Entferne die Konstante:

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

= 6 \cdot \int \frac{1}{2 x + 3 x ^{2}} d x

Ermittle den Partialbruch von

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

= 6 \cdot \int \frac{1}{2 x} - \frac{3}{2 ( 3 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

= 6 (\int \frac{1}{2 x} d x - \int \frac{3}{2 ( 3 x + 2 )} d x)

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{3}{2 ( 3 x + 2 )} d x = \frac{1}{2} ln ∣ 3 x + 2 ∣

= 6 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ 3 x + 2 ∣)

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ 3 x + 2 ∣) + C

Graph

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