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Beliebt Rechnen >

integral of (5x^2+3x+5)/((x^2+1)^2)

Lösung

\int \frac{5 x ^{2} + 3 x + 5}{( x ^{2} + 1 ) ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{5 x ^{2} + 3 x + 5}{( x ^{2} + 1 ) ^{2}} d x

Ermittle den Partialbruch von

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

= \int \frac{5}{x ^{2} + 1} + \frac{3 x}{( x ^{2} + 1 ) ^{2}} d x

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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