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Beliebt Rechnen >

integral of (x-4)/(x(x-2)(x-5))

Lösung

\int \frac{x - 4}{x ( x - 2 ) ( x - 5 )} d x

Lösung

\frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x - 4}{x ( x - 2 ) ( x - 5 )} d x

Multipliziere aus

\frac{x - 4}{x ( x - 2 ) ( x - 5 )} : \frac{x}{x ( x - 2 ) ( x - 5 )} - \frac{4}{x ( x - 2 ) ( 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{x}{x ( x - 2 ) ( x - 5 )} d x - \int \frac{4}{x ( x - 2 ) ( x - 5 )} d x

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

\int \frac{4}{x ( x - 2 ) ( x - 5 )} d x = 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣)

= - \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 5 ∣ - 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣)

Vereinfache

- \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 5 ∣ - 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣) : \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣

= \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of (40)/(S^2)

\int \frac{40}{S ^{2}} d S

integral of ln(2x-3)

\int ln (2 x - 3) d x

integral of (x^3-x^2-x+1)e^{2x}

\int (x^{3} - x^{2} - x + 1) e^{2 x} d x

integral of 1/(sqrt(4x^2+8x+19))

\int \frac{1}{4 x ^{2} + 8 x + 19} d x

integral of ln^2((3a)/x)

\int ln^{2} (\frac{3 a}{x}) d x