integral 1/((a^3-2a))

Lösung

integral

\frac{1}{( a ^{3} - 2 a )}

- \frac{1}{2} ln ∣ a ∣ + \frac{1}{4} ln a + 2 + \frac{1}{4} ln a - 2 + C

Schritte zur Lösung

\int \frac{1}{a ^{3} - 2 a} d a

Ermittle den Partialbruch von

\frac{1}{a ^{3} - 2 a} : - \frac{1}{2 a} + \frac{1}{4 ( a + 2 )} + \frac{1}{4 ( a - 2 )}

= \int - \frac{1}{2 a} + \frac{1}{4 ( a + 2 )} + \frac{1}{4 ( a - 2 )} d a

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}{2 a} d a + \int \frac{1}{4 ( a + 2 )} d a + \int \frac{1}{4 ( a - 2 )} d a

\int \frac{1}{2 a} d a = \frac{1}{2} ln ∣ a ∣

\int \frac{1}{4 ( a + 2 )} d a = \frac{1}{4} ln a + 2

\int \frac{1}{4 ( a - 2 )} d a = \frac{1}{4} ln a - 2

= - \frac{1}{2} ln ∣ a ∣ + \frac{1}{4} ln a + 2 + \frac{1}{4} ln a - 2

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ a ∣ + \frac{1}{4} ln a + 2 + \frac{1}{4} ln a - 2 + C

in t e g r a l (9 x + 4)^{4}

in t e g r a l cos (2 x + 3)

in t e g r a l 2 x 2 x - 3

in t e g r a l \frac{1}{[ ( 1 - x ^{2} ) ( 1 + x ) ]}

in t e g r a l 7 x^{6}