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Beliebt Rechnen >

integral of (x^3)/(4sqrt(4+x^2))

Lösung

\int \frac{x ^{3}}{4 4 + x ^{2}} d x

Lösung

\frac{1}{12} (4 + x^{2})^{\frac{3}{2}} - 4 + x^{2} + C

Schritte zur Lösung

\int \frac{x ^{3}}{4 4 + x ^{2}} d x

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{x ^{3}}{4 + x ^{2}} d x

Wende U-Substitution an

= \frac{1}{4} \cdot \int u^{2} - 4 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}{4} (\int u^{2} d u - \int 4 d u)

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

\int 4 d u = 4 u

= \frac{1}{4} (\frac{u ^{3}}{3} - 4 u)

Setze in

u = 4 + x^{2}

ein

= \frac{1}{4} (\frac{( 4 + x ^{2} ) ^{3}}{3} - 4 4 + x^{2})

Vereinfache

\frac{1}{4} (\frac{( 4 + x ^{2} ) ^{3}}{3} - 4 4 + x^{2}) : \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} - 4 + x^{2}

= \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} - 4 + x^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} - 4 + x^{2} + C

Graph

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