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Beliebt Rechnen >

integral of (6x+8)/(3x^2+8x-2)

Lösung

\int \frac{6 x + 8}{3 x ^{2} + 8 x - 2} d x

Lösung

ln 9 x^{2} + 24 x - 6 + C

Schritte zur Lösung

\int \frac{6 x + 8}{3 x ^{2} + 8 x - 2} d x

Multipliziere aus

\frac{6 x + 8}{3 x ^{2} + 8 x - 2} : \frac{6 x}{3 x ^{2} + 8 x - 2} + \frac{8}{3 x ^{2} + 8 x - 2}

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{6 x}{3 x ^{2} + 8 x - 2} d x + \int \frac{8}{3 x ^{2} + 8 x - 2} d x

\int \frac{6 x}{3 x ^{2} + 8 x - 2} d x = ln 9 x^{2} + 24 x - 6 + \frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1)

\int \frac{8}{3 x ^{2} + 8 x - 2} d x = - \frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1)

= ln 9 x^{2} + 24 x - 6 + \frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1) - \frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1)

Addiere gleiche Elemente:

\frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1) - \frac{2 2}{11} (ln \frac{3 x + 4}{22} + 1 - ln \frac{3 x + 4}{22} - 1) = 0

= ln 9 x^{2} + 24 x - 6

Füge eine Konstante zur Lösung hinzu

= ln 9 x^{2} + 24 x - 6 + C

Graph

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