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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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