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-1)/((x^2+4x+5)^2)

Lösung

\int \frac{x - 1}{( x ^{2} + 4 x + 5 ) ^{2}} d x

Lösung

- \frac{1}{2 ( x ^{2} + 4 x + 5 )} - \frac{3 ( arctan ( x + 2 ) ( x ^{2} + 4 x + 5 ) + x + 2 )}{2 ( x ^{2} + 4 x + 5 )} + C

Schritte zur Lösung

\int \frac{x - 1}{( x ^{2} + 4 x + 5 ) ^{2}} d x

Vervollständige das Quadrat

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

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

Wende U-Substitution an

= \int \frac{u - 3}{( u ^{2} + 1 ) ^{2}} d u

Multipliziere aus

\frac{u - 3}{( u ^{2} + 1 ) ^{2}} : \frac{u}{( u ^{2} + 1 ) ^{2}} - \frac{3}{( u ^{2} + 1 ) ^{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{u}{( u ^{2} + 1 ) ^{2}} d u - \int \frac{3}{( u ^{2} + 1 ) ^{2}} d u

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

\int \frac{3}{( u ^{2} + 1 ) ^{2}} d u = \frac{3}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

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

Setze in

u = x + 2

ein

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

Vereinfache

- \frac{1}{2 ( ( x + 2 ) ^{2} + 1 )} - \frac{3}{2} (arctan (x + 2) + \frac{x + 2}{1 + ( x + 2 ) ^{2}}) : - \frac{1}{2 ( x ^{2} + 4 x + 5 )} - \frac{3 ( arctan ( x + 2 ) ( x ^{2} + 4 x + 5 ) + x + 2 )}{2 ( x ^{2} + 4 x + 5 )}

= - \frac{1}{2 ( x ^{2} + 4 x + 5 )} - \frac{3 ( arctan ( x + 2 ) ( x ^{2} + 4 x + 5 ) + x + 2 )}{2 ( x ^{2} + 4 x + 5 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x ^{2} + 4 x + 5 )} - \frac{3 ( arctan ( x + 2 ) ( x ^{2} + 4 x + 5 ) + x + 2 )}{2 ( x ^{2} + 4 x + 5 )} + C

Graph

Beliebte Beispiele

limit as x approaches infinity of (-2)/x

x \to \infty lim (\frac{- 2}{x})

integral of 4sin^2(5x+3)+10x^{2/3}-1

\int 4 sin^{2} (5 x + 3) + 10 x^{\frac{2}{3}} - 1 d x

derivative of (x+1/(3x+2))

\frac{d}{d x} (\frac{x + 1}{3 x + 2})

erweitern 6x(x^2-5)^2

e x p an d 6 x (x^{2} - 5)^{2}

derivative of (9x/(x+4))

\frac{d}{d x} (\frac{9 x}{x + 4})