integral of (4x^2+4)/((x^2-2x+2)^2)

Lösung

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

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

Faktorisiere

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

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \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

= 4 (\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)))

= 4 (\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

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

\frac{\partial}{\partial y} (ln (x^{2} + y^{2} + 1))

\frac{\partial}{\partial x} (3 - \frac{e ^{y}}{e ^{x + 1}})

d er i v a t i v e f (x) = 7 x^{6}

t an g e n t f (x) = 2 x^{2} + 3 x - 2, at x = 0

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