integral 1/((x(x+2)))

Lösung

integral

\frac{1}{( x ( x + 2 ) )}

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{x ( x + 2 )} : \frac{1}{2 x} - \frac{1}{2 ( x + 2 )}

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 5^{- x}

in t e g r a l \frac{y}{( x ^{2} + y ^{2} )}

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

in t e g r a l 1 + y^{2}

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