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Beliebt Rechnen >

integral of (x^3)/(sqrt(x^2+16))

Lösung

\int \frac{x ^{3}}{x ^{2} + 16} d x

Lösung

\frac{1}{3} (x^{2} + 16)^{\frac{3}{2}} - 16 x^{2} + 16 + C

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 16} d x

Wende U-Substitution an

= \int u^{2} - 16 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 16 d u

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

\int 16 d u = 16 u

= \frac{u ^{3}}{3} - 16 u

Setze in

u = x^{2} + 16

ein

= \frac{( x ^{2} + 16 ) ^{3}}{3} - 16 x^{2} + 16

Vereinfache

\frac{( x ^{2} + 16 ) ^{3}}{3} - 16 x^{2} + 16 : \frac{1}{3} (x^{2} + 16)^{\frac{3}{2}} - 16 x^{2} + 16

= \frac{1}{3} (x^{2} + 16)^{\frac{3}{2}} - 16 x^{2} + 16

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (x^{2} + 16)^{\frac{3}{2}} - 16 x^{2} + 16 + C

Graph

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