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

Lösung

integral

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

ln ∣ x + 1 ∣ + \frac{2}{x + 1} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

= ln ∣ x + 1 ∣ + \frac{2}{x + 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 1 ∣ + \frac{2}{x + 1} + C

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

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

in t e g r a l \frac{1}{1 + x}

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

in t e g r a l x + 3