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

Lösung

integral

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int 1 d x = x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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

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

in t e g r a l 5 + x

in t e g r a l (2 x + 1)^{3}

in t e g r a l 2 + \frac{1}{x ^{2}} - \frac{3}{x ^{4}}