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Beliebt Rechnen >

integral of (5x^3)/(sqrt(6+x^2))

Lösung

\int \frac{5 x ^{3}}{6 + x ^{2}} d x

Lösung

5 (\frac{1}{3} (6 + x^{2})^{\frac{3}{2}} - 6 6 + x^{2}) + C

Schritte zur Lösung

\int \frac{5 x ^{3}}{6 + x ^{2}} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{x ^{3}}{6 + x ^{2}} d x

Wende U-Substitution an

= 5 \cdot \int u^{2} - 6 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

= 5 (\int u^{2} d u - \int 6 d u)

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

\int 6 d u = 6 u

= 5 (\frac{u ^{3}}{3} - 6 u)

Setze in

u = 6 + x^{2}

ein

= 5 (\frac{( 6 + x ^{2} ) ^{3}}{3} - 6 6 + x^{2})

Vereinfache

5 (\frac{( 6 + x ^{2} ) ^{3}}{3} - 6 6 + x^{2}) : 5 (\frac{1}{3} (6 + x^{2})^{\frac{3}{2}} - 6 6 + x^{2})

= 5 (\frac{1}{3} (6 + x^{2})^{\frac{3}{2}} - 6 6 + x^{2})

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{1}{3} (6 + x^{2})^{\frac{3}{2}} - 6 6 + x^{2}) + C

Graph

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