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Beliebt Rechnen >

integral of (2x^2+6)/((x^2-2x+2)^2)

Lösung

\int \frac{2 x ^{2} + 6}{( x ^{2} - 2 x + 2 ) ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{2 x ^{2} + 6}{( x ^{2} - 2 x + 2 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{2 x ^{2} + 6}{( x ^{2} - 2 x + 2 ) ^{2}} : \frac{2}{x ^{2} - 2 x + 2} + \frac{4 x + 2}{( x ^{2} - 2 x + 2 ) ^{2}}

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches infinity of ln(2x)

x \to \infty lim (ln (2 x))

fläche x=-1,x=3,y=x^3-1,y=0

a re a x = - 1, x = 3, y = x^{3} - 1, y = 0

derivative of x^8ln(5x)

\frac{d}{d x} (x^{8} ln (5 x))

derivative of 1200sqrt(x^2-0.2x)

\frac{d}{d x} (1200 x^{2} - 0.2 x)

derivative of x^3-12x-5

\frac{d}{d x} (x^{3} - 12 x - 5)