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Beliebt Rechnen >

integral of (x^3)/(sqrt(5+x^2))

Lösung

\int \frac{x ^{3}}{5 + x ^{2}} d x

Lösung

\frac{1}{3} (5 + x^{2})^{\frac{3}{2}} - 5 5 + x^{2} + C

Schritte zur Lösung

\int \frac{x ^{3}}{5 + x ^{2}} d x

Wende U-Substitution an

= \int u^{2} - 5 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

= \int u^{2} d u - \int 5 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 5 d u = 5 u

= \frac{u ^{3}}{3} - 5 u

Setze in

u = 5 + x^{2}

ein

= \frac{( 5 + x ^{2} ) ^{3}}{3} - 5 5 + x^{2}

Vereinfache

\frac{( 5 + x ^{2} ) ^{3}}{3} - 5 5 + x^{2} : \frac{1}{3} (5 + x^{2})^{\frac{3}{2}} - 5 5 + x^{2}

= \frac{1}{3} (5 + x^{2})^{\frac{3}{2}} - 5 5 + x^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (5 + x^{2})^{\frac{3}{2}} - 5 5 + x^{2} + C

Graph

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