integral of (x+3)/(x^4+9x^2)

Lösung

\int \frac{x + 3}{x ^{4} + 9 x ^{2}} d x

\frac{1}{9} ln ∣ x ∣ - \frac{1}{3 x} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3})) + C

Schritte zur Lösung

\int \frac{x + 3}{x ^{4} + 9 x ^{2}} d x

Ermittle den Partialbruch von

\frac{x + 3}{x ^{4} + 9 x ^{2}} : \frac{1}{9 x} + \frac{1}{3 x ^{2}} + \frac{- x - 3}{9 ( x ^{2} + 9 )}

= \int \frac{1}{9 x} + \frac{1}{3 x ^{2}} + \frac{- x - 3}{9 ( x ^{2} + 9 )} 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}{3 x ^{2}} d x + \int \frac{- x - 3}{9 ( x ^{2} + 9 )} d x

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

\int \frac{1}{3 x ^{2}} d x = - \frac{1}{3 x}

\int \frac{- x - 3}{9 ( x ^{2} + 9 )} d x = \frac{1}{9} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}))

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{3 x} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{3 x} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 9 - arctan (\frac{x}{3})) + C