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

Lösung

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

- \frac{x + 1}{2 ( x ^{2} + 2 x + 2 )} - \frac{3}{2} arctan (x + 1) - \frac{1}{2} sin (2 arctan (x + 1)) - \frac{1}{2 ( x ^{2} + 2 x + 2 )} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

= - \frac{x + 1}{2 ( x ^{2} + 2 x + 2 )} - \frac{3}{2} arctan (x + 1) - \frac{1}{2} sin (2 arctan (x + 1)) - \frac{1}{2 ( x ^{2} + 2 x + 2 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{x + 1}{2 ( x ^{2} + 2 x + 2 )} - \frac{3}{2} arctan (x + 1) - \frac{1}{2} sin (2 arctan (x + 1)) - \frac{1}{2 ( x ^{2} + 2 x + 2 )} + C

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

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