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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \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

= \int u^{2} d u - \int 9 d u

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

\int 9 d u = 9 u

= \frac{u ^{3}}{3} - 9 u

Setze in

u = 9 - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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