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Beliebt Rechnen >

integral of (2x-1)/(x^2+4x-5)

Lösung

\int \frac{2 x - 1}{x ^{2} + 4 x - 5} d x

Lösung

\frac{1}{6} ln ∣ x - 1 ∣ + \frac{11}{6} ln ∣ x + 5 ∣ + C

Schritte zur Lösung

\int \frac{2 x - 1}{x ^{2} + 4 x - 5} d x

Multipliziere aus

\frac{2 x - 1}{x ^{2} + 4 x - 5} : \frac{2 x}{x ^{2} + 4 x - 5} - \frac{1}{x ^{2} + 4 x - 5}

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}{x ^{2} + 4 x - 5} d x - \int \frac{1}{x ^{2} + 4 x - 5} d x

\int \frac{2 x}{x ^{2} + 4 x - 5} d x = 2 (\frac{1}{6} ln ∣ x - 1 ∣ + \frac{5}{6} ln ∣ x + 5 ∣)

\int \frac{1}{x ^{2} + 4 x - 5} d x = \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6} ln ∣ x + 5 ∣

= 2 (\frac{1}{6} ln ∣ x - 1 ∣ + \frac{5}{6} ln ∣ x + 5 ∣) - (\frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6} ln ∣ x + 5 ∣)

Vereinfache

2 (\frac{1}{6} ln ∣ x - 1 ∣ + \frac{5}{6} ln ∣ x + 5 ∣) - (\frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6} ln ∣ x + 5 ∣) : \frac{1}{6} ln ∣ x - 1 ∣ + \frac{11}{6} ln ∣ x + 5 ∣

= \frac{1}{6} ln ∣ x - 1 ∣ + \frac{11}{6} ln ∣ x + 5 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} ln ∣ x - 1 ∣ + \frac{11}{6} ln ∣ x + 5 ∣ + C

Graph

Beliebte Beispiele

integral of 3/(2-5x)

\int \frac{3}{2 - 5 x} d x

integral of (p^2-9p)/(\sqrt[5]{p)}

\int \frac{p ^{2} - 9 p}{5 p} d p

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

\int \frac{3 x + 4}{x ( x + 2 ) ^{2}} d x

integral of (t^4+4)/(t^4)

\int \frac{t ^{4} + 4}{t ^{4}} d t

integral of sin^3(x)cos^2(y)

\int sin^{3} (x) cos^{2} (y) d x