integral of (x^3)/((x^2-2x+2)^2)

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1 + x}{3 x} d x

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