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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{5}{21} ln ∣ x + 3 ∣ + \frac{1}{14} ln ∣ x - 4 ∣ + \frac{1}{6} ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x}{x ( x + 3 ) ( x - 4 )} d x = - \frac{1}{7} ln ∣ x + 3 ∣ + \frac{1}{7} ln ∣ x - 4 ∣

\int \frac{2}{x ( x + 3 ) ( x - 4 )} d x = 2 (\frac{1}{21} ln ∣ x + 3 ∣ - \frac{1}{12} ln ∣ x ∣ + \frac{1}{28} ln ∣ x - 4 ∣)

= - \frac{1}{7} ln ∣ x + 3 ∣ + \frac{1}{7} ln ∣ x - 4 ∣ - 2 (\frac{1}{21} ln ∣ x + 3 ∣ - \frac{1}{12} ln ∣ x ∣ + \frac{1}{28} ln ∣ x - 4 ∣)

Vereinfache

- \frac{1}{7} ln ∣ x + 3 ∣ + \frac{1}{7} ln ∣ x - 4 ∣ - 2 (\frac{1}{21} ln ∣ x + 3 ∣ - \frac{1}{12} ln ∣ x ∣ + \frac{1}{28} ln ∣ x - 4 ∣) : - \frac{5}{21} ln ∣ x + 3 ∣ + \frac{1}{14} ln ∣ x - 4 ∣ + \frac{1}{6} ln ∣ x ∣

= - \frac{5}{21} ln ∣ x + 3 ∣ + \frac{1}{14} ln ∣ x - 4 ∣ + \frac{1}{6} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{21} ln ∣ x + 3 ∣ + \frac{1}{14} ln ∣ x - 4 ∣ + \frac{1}{6} ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of (x^2)/(1+sqrt(x))

\int \frac{x ^{2}}{1 + x} d x

integral of (3y^2+1)ln(y^2-1)

\int (3 y^{2} + 1) ln (y^{2} - 1) d y

integral of (cos^3(y))/(1-sin(y))

\int \frac{cos ^{3} ( y )}{1 - sin ( y )} d y

integral of (x-2)/(x(x-1))

\int \frac{x - 2}{x ( x - 1 )} d x

integral of x(4-x^2)

\int x (4 - x^{2}) d x