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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x}{1 + x ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Vereinfache

= \frac{x}{1 + x ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{x}{1 + x ^{2}} + C

Graph

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