integral of (3x-1)/(x^2-3x-4)

Lösung

\int \frac{3 x - 1}{x ^{2} - 3 x - 4} d x

\frac{4}{5} ln ∣ x + 1 ∣ + \frac{11}{5} ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{3 x - 1}{x ^{2} - 3 x - 4} d x

Multipliziere aus

\frac{3 x - 1}{x ^{2} - 3 x - 4} : \frac{3 x}{x ^{2} - 3 x - 4} - \frac{1}{x ^{2} - 3 x - 4}

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{3 x}{x ^{2} - 3 x - 4} d x - \int \frac{1}{x ^{2} - 3 x - 4} d x

\int \frac{3 x}{x ^{2} - 3 x - 4} d x = 3 (\frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣)

\int \frac{1}{x ^{2} - 3 x - 4} d x = - \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x - 4 ∣

= 3 (\frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣) - (- \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x - 4 ∣)

Vereinfache

3 (\frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣) - (- \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x - 4 ∣) : \frac{4}{5} ln ∣ x + 1 ∣ + \frac{11}{5} ln ∣ x - 4 ∣

= \frac{4}{5} ln ∣ x + 1 ∣ + \frac{11}{5} ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{5} ln ∣ x + 1 ∣ + \frac{11}{5} ln ∣ x - 4 ∣ + C