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Beliebt Rechnen >

integral of (4x^2)/((x^2+1)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Multipliziere aus

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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