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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{( x ^{2} - 1 ) ^{2}}{( x ^{2} + 1 ) ^{2}} : 1 - \frac{4}{x ^{2} + 1} + \frac{4}{( x ^{2} + 1 ) ^{2}}

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

\int 1 d x = x

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

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

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

Vereinfache

x - 4 arctan (x) + 2 (arctan (x) + \frac{x}{1 + x ^{2}}) : x - 2 arctan (x) + \frac{2 x}{1 + x ^{2}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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