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

Lösung

integral

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

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{3} - x - 2}{1 - x ^{2}} : \frac{x ^{3}}{1 - x ^{2}} - \frac{x}{1 - x ^{2}} - \frac{2}{1 - x ^{2}}

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l x^{3} - 3 x^{2} + 5 x + 2

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

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

s im pl i f y (\frac{1}{x - 1})^{\frac{1}{2}}

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