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

Lösung

\int (\frac{x ^{3}}{16 - x ^{2}}) d x

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

Schritte zur Lösung

\int \frac{x ^{3}}{16 - x ^{2}} 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 = 16 - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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