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Beliebt Rechnen >

integral of 1/(x(9-x^2))

Lösung

\int \frac{1}{x ( 9 - x ^{2} )} d x

Lösung

\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{18} ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ( 9 - x ^{2} )} d x

Ermittle den Partialbruch von

\frac{1}{x ( 9 - x ^{2} )} : \frac{1}{9 x} - \frac{1}{18 ( x + 3 )} - \frac{1}{18 ( x - 3 )}

= \int \frac{1}{9 x} - \frac{1}{18 ( x + 3 )} - \frac{1}{18 ( x - 3 )} d x

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{1}{9 x} d x - \int \frac{1}{18 ( x + 3 )} d x - \int \frac{1}{18 ( x - 3 )} d x

\int \frac{1}{9 x} d x = \frac{1}{9} ln ∣ x ∣

\int \frac{1}{18 ( x + 3 )} d x = \frac{1}{18} ln ∣ x + 3 ∣

\int \frac{1}{18 ( x - 3 )} d x = \frac{1}{18} ln ∣ x - 3 ∣

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{18} ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln ∣ x + 3 ∣ - \frac{1}{18} ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

integral of in^3x

\int i n^{3} x d x

integral of-30e^{12t}sin(t)

\int - 30 e^{12 t} sin (t) d t

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

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

integral of e^{4x^2}8x

\int e^{4 x^{2}} 8 x d x

integral of (cos(x/2)+sin(x/2))^2

\int (cos (\frac{x}{2}) + sin (\frac{x}{2}))^{2} d x