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Beliebt Rechnen >

integral of (-4x^3)/(sqrt(x^2+9))

Lösung

\int \frac{- 4 x ^{3}}{x ^{2} + 9} d x

Lösung

- 4 (\frac{1}{3} (x^{2} + 9)^{\frac{3}{2}} - 9 x^{2} + 9) + C

Schritte zur Lösung

\int \frac{- 4 x ^{3}}{x ^{2} + 9} d x

Entferne die Konstante:

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

= - 4 \cdot \int \frac{x ^{3}}{x ^{2} + 9} d x

Wende U-Substitution an

= - 4 \cdot \int u^{2} - 9 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

= - 4 (\int u^{2} d u - \int 9 d u)

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

\int 9 d u = 9 u

= - 4 (\frac{u ^{3}}{3} - 9 u)

Setze in

u = x^{2} + 9

ein

= - 4 (\frac{( x ^{2} + 9 ) ^{3}}{3} - 9 x^{2} + 9)

Vereinfache

- 4 (\frac{( x ^{2} + 9 ) ^{3}}{3} - 9 x^{2} + 9) : - 4 (\frac{1}{3} (x^{2} + 9)^{\frac{3}{2}} - 9 x^{2} + 9)

= - 4 (\frac{1}{3} (x^{2} + 9)^{\frac{3}{2}} - 9 x^{2} + 9)

Füge eine Konstante zur Lösung hinzu

= - 4 (\frac{1}{3} (x^{2} + 9)^{\frac{3}{2}} - 9 x^{2} + 9) + C

Graph

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\int 12 x^{3} - 4 x d x

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