integral of 1/(x^3(x^2+1)^2)

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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