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

Lösung

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

- \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{8} ln ∣ x - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{- 2 x ^{4} arctan ( x ) - 2 x ^{3} + x ^{2} - 4 x ^{2} arctan ( x ) - 2 x - 2 arctan ( x ) + 1}{4 ( x ^{2} + 1 ) ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x + 2}{( x ^{2} - 1 ) ( x ^{2} + 1 ) ^{2}} : - \frac{1}{8 ( x + 1 )} + \frac{3}{8 ( x - 1 )} + \frac{- x - 2}{4 ( x ^{2} + 1 )} + \frac{- x - 2}{2 ( x ^{2} + 1 ) ^{2}}

= \int - \frac{1}{8 ( x + 1 )} + \frac{3}{8 ( x - 1 )} + \frac{- x - 2}{4 ( x ^{2} + 1 )} + \frac{- x - 2}{2 ( 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{1}{8 ( x + 1 )} d x + \int \frac{3}{8 ( x - 1 )} d x + \int \frac{- x - 2}{4 ( x ^{2} + 1 )} d x + \int \frac{- x - 2}{2 ( x ^{2} + 1 ) ^{2}} d x

\int \frac{1}{8 ( x + 1 )} d x = \frac{1}{8} ln ∣ x + 1 ∣

\int \frac{3}{8 ( x - 1 )} d x = \frac{3}{8} ln ∣ x - 1 ∣

\int \frac{- x - 2}{4 ( x ^{2} + 1 )} d x = \frac{1}{4} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x))

\int \frac{- x - 2}{2 ( x ^{2} + 1 ) ^{2}} d x = \frac{- 2 x ^{4} arctan ( x ) - 2 x ^{3} + x ^{2} - 4 x ^{2} arctan ( x ) - 2 x - 2 arctan ( x ) + 1}{4 ( x ^{2} + 1 ) ^{2}}

= - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{8} ln ∣ x - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{- 2 x ^{4} arctan ( x ) - 2 x ^{3} + x ^{2} - 4 x ^{2} arctan ( x ) - 2 x - 2 arctan ( x ) + 1}{4 ( x ^{2} + 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{8} ln ∣ x - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{- 2 x ^{4} arctan ( x ) - 2 x ^{3} + x ^{2} - 4 x ^{2} arctan ( x ) - 2 x - 2 arctan ( x ) + 1}{4 ( x ^{2} + 1 ) ^{2}} + C

\int \frac{( x - 2 ) ^{2}}{x ^{4}} d x

\int \frac{( 12 x + 1 )}{4 x - 3} d x

\int 3 x^{2} - 3 x (2 x - 3) d x

\int \frac{1}{( x ^{2} + 2 x + 10 ) ^{2}} d x

\int \frac{4 x ^{3}}{( x ^{2} + 1 ) ^{3}} d x