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Beliebt Rechnen >

integral of (x^3)/((a^2+x^2)^2)

Lösung

\int \frac{x ^{3}}{( a ^{2} + x ^{2} ) ^{2}} d x

Lösung

\frac{1}{2} (ln a^{2} + x^{2} + \frac{a ^{2}}{a ^{2} + x ^{2}}) + C

Schritte zur Lösung

\int \frac{x ^{3}}{( a ^{2} + x ^{2} ) ^{2}} d x

Wende U-Substitution an

= \int \frac{u - a ^{2}}{2 u ^{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 - a ^{2}}{u ^{2}} d u

Multipliziere aus

\frac{u - a ^{2}}{u ^{2}} : \frac{1}{u} - \frac{a ^{2}}{u ^{2}}

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{a ^{2}}{u ^{2}} d u = - \frac{a ^{2}}{u}

= \frac{1}{2} (ln ∣ u ∣ - (- \frac{a ^{2}}{u}))

Setze in

u = a^{2} + x^{2}

ein

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

Vereinfache

= \frac{1}{2} (ln a^{2} + x^{2} + \frac{a ^{2}}{a ^{2} + x ^{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (ln a^{2} + x^{2} + \frac{a ^{2}}{a ^{2} + x ^{2}}) + C

Graph

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