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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x ^{2}}{( 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} sin (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}

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

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

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

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

d er i v a t i v e \frac{x - 13}{x - 13}

(6 x + 1) y^{2} \frac{d y}{d x} + 4 x^{2} + 2 y^{3} = 0

\frac{\partial}{\partial x} (y x^{2} z^{2} - x y^{4} + x z)

\frac{d}{d x} (ln (x) cos (x))

x^{2} y^{^{''}} + 3 x y^{^{'}} - 15 y = 0