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Beliebt Rechnen >

integral of ((3x+5))/((x^2+2x+2)^2)

Lösung

\int \frac{( 3 x + 5 )}{( x ^{2} + 2 x + 2 ) ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{3 x + 5}{( x ^{2} + 2 x + 2 ) ^{2}} d x

Multipliziere aus

\frac{3 x + 5}{( x ^{2} + 2 x + 2 ) ^{2}} : \frac{3 x}{( x ^{2} + 2 x + 2 ) ^{2}} + \frac{5}{( x ^{2} + 2 x + 2 ) ^{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{3 x}{( x ^{2} + 2 x + 2 ) ^{2}} d x + \int \frac{5}{( x ^{2} + 2 x + 2 ) ^{2}} d x

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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