de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

\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 )} + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} - 2 x + 2 : (x - 1)^{2} + 1

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

Wende U-Substitution an

= \int \frac{u + 1}{( u ^{2} + 1 ) ^{2}} d u

Wende die partielle Integration an

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

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

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

Setze in

u = x - 1

ein

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

Vereinfache

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

= \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 )}

Füge eine Konstante zur Lösung hinzu

= \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 )} + C

Graph

Beliebte Beispiele

integral of (e^{6x})/(1+e^{6x)}

\int \frac{e ^{6 x}}{1 + e ^{6 x}} d x

integral of ze^{yz}

\int z e^{yz} d y

integral of 12x^2(4x^3-3)

\int 12 x^{2} (4 x^{3} - 3) d x

integral of cos(x)in(x)

\int cos (x) d sin (x) d x

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

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