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

Lösung

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

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

Schritte zur Lösung

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Setze in

u = \frac{x}{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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