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

Lösung

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

- \frac{1}{2 ( x ^{2} + 2 x + 4 )} - \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1)))) + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} + 2 x + 4 : (x + 1)^{2} + 3

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

Wende U-Substitution an

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

Multipliziere aus

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

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

\int \frac{1}{( u ^{2} + 3 ) ^{2}} d u = \frac{1}{6 3} (arctan (\frac{u}{3}) + \frac{1}{2} sin (2 arctan (\frac{u}{3})))

= - \frac{1}{2 ( u ^{2} + 3 )} - \frac{1}{6 3} (arctan (\frac{u}{3}) + \frac{1}{2} sin (2 arctan (\frac{u}{3})))

Setze in

u = x + 1

ein

= - \frac{1}{2 ( ( x + 1 ) ^{2} + 3 )} - \frac{1}{6 3} (arctan (\frac{x + 1}{3}) + \frac{1}{2} sin (2 arctan (\frac{x + 1}{3})))

Vereinfache

- \frac{1}{2 ( ( x + 1 ) ^{2} + 3 )} - \frac{1}{6 3} (arctan (\frac{x + 1}{3}) + \frac{1}{2} sin (2 arctan (\frac{x + 1}{3}))) : - \frac{1}{2 ( x ^{2} + 2 x + 4 )} - \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1))))

= - \frac{1}{2 ( x ^{2} + 2 x + 4 )} - \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1))))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x ^{2} + 2 x + 4 )} - \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1)))) + C

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