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Beliebt Rechnen >

integral of (x+3)/((x^2+2x+3)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Addiere gleiche Elemente:

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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