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Beliebt Rechnen >

integral of ((2x+8))/((x^2+2x+5)^2)

Lösung

\int \frac{( 2 x + 8 )}{( x ^{2} + 2 x + 5 ) ^{2}} d x

Lösung

\frac{3}{8} arctan (\frac{1}{2} (x + 1)) + \frac{3}{16} sin (2 arctan (\frac{1}{2} (x + 1))) - \frac{1}{x ^{2} + 2 x + 5} + C

Schritte zur Lösung

\int \frac{2 x + 8}{( x ^{2} + 2 x + 5 ) ^{2}} d x

Multipliziere aus

\frac{2 x + 8}{( x ^{2} + 2 x + 5 ) ^{2}} : \frac{2 x}{( x ^{2} + 2 x + 5 ) ^{2}} + \frac{8}{( x ^{2} + 2 x + 5 ) ^{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{2 x}{( x ^{2} + 2 x + 5 ) ^{2}} d x + \int \frac{8}{( x ^{2} + 2 x + 5 ) ^{2}} d x

\int \frac{2 x}{( x ^{2} + 2 x + 5 ) ^{2}} d x = - \frac{1}{x ^{2} + 2 x + 5} - \frac{1}{8} arctan (\frac{1}{2} (x + 1)) - \frac{1}{16} sin (2 arctan (\frac{1}{2} (x + 1)))

\int \frac{8}{( x ^{2} + 2 x + 5 ) ^{2}} d x = \frac{1}{2} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1))))

= - \frac{1}{x ^{2} + 2 x + 5} - \frac{1}{8} arctan (\frac{1}{2} (x + 1)) - \frac{1}{16} sin (2 arctan (\frac{1}{2} (x + 1))) + \frac{1}{2} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1))))

Vereinfache

- \frac{1}{x ^{2} + 2 x + 5} - \frac{1}{8} arctan (\frac{1}{2} (x + 1)) - \frac{1}{16} sin (2 arctan (\frac{1}{2} (x + 1))) + \frac{1}{2} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1)))) : \frac{3}{8} arctan (\frac{1}{2} (x + 1)) + \frac{3}{16} sin (2 arctan (\frac{1}{2} (x + 1))) - \frac{1}{x ^{2} + 2 x + 5}

= \frac{3}{8} arctan (\frac{1}{2} (x + 1)) + \frac{3}{16} sin (2 arctan (\frac{1}{2} (x + 1))) - \frac{1}{x ^{2} + 2 x + 5}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{8} arctan (\frac{1}{2} (x + 1)) + \frac{3}{16} sin (2 arctan (\frac{1}{2} (x + 1))) - \frac{1}{x ^{2} + 2 x + 5} + C

Graph

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