integral of (2x-3)/(3x^2+4x-7)

Lösung

\int \frac{2 x - 3}{3 x ^{2} + 4 x - 7} d x

2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) + \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x) + C

Schritte zur Lösung

\int \frac{2 x - 3}{3 x ^{2} + 4 x - 7} d x

Multipliziere aus

\frac{2 x - 3}{3 x ^{2} + 4 x - 7} : \frac{2 x}{3 x ^{2} + 4 x - 7} - \frac{3}{3 x ^{2} + 4 x - 7}

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{2 x}{3 x ^{2} + 4 x - 7} d x - \int \frac{3}{3 x ^{2} + 4 x - 7} d x

\int \frac{2 x}{3 x ^{2} + 4 x - 7} d x = 2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x))

\int \frac{3}{3 x ^{2} + 4 x - 7} d x = - \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)

= 2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) - (- \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x))

Vereinfache

2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) - (- \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) : 2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) + \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)

= 2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) + \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{6} ln 9 x^{2} + 12 x - 21 + \frac{1}{15} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x)) + \frac{3}{10} (ln \frac{7}{5} + \frac{3}{5} x - ln - \frac{3}{5} + \frac{3}{5} x) + C

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