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

Lösung

\int (9 - x^{2})^{3} d x

x (9 - x^{2})^{\frac{3}{2}} + \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) + C

Schritte zur Lösung

\int (9 - x^{2})^{3} d x

Wende die partielle Integration an

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

\int - 3 x^{2} 9 - x^{2} d x = - \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x)))

= x (9 - x^{2})^{3} - (- \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))))

Vereinfache

x (9 - x^{2})^{3} - (- \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x)))) : x (9 - x^{2})^{\frac{3}{2}} + \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x)))

= x (9 - x^{2})^{\frac{3}{2}} + \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x)))

Füge eine Konstante zur Lösung hinzu

= x (9 - x^{2})^{\frac{3}{2}} + \frac{243}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) + C

\int x^{2} (3 x^{3} - 1)^{4} d x

\int cos (5 x) sin (5 x) d x

\int 17 x d x

\int sin (\frac{π t}{3}) d t

\int \frac{x ^{3} - 6}{x ^{4} + 6 x ^{2} + 8} d x