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

Lösung

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

ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C