integral of x^4arcsin(x)

Lösung

\int x^{4} arcsin (x) d x

\frac{1}{5} x^{5} arcsin (x) - \frac{1}{5} (- 1 - x^{2} + \frac{2}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}}) + C

Schritte zur Lösung

\int x^{4} arcsin (x) d x

Wende die partielle Integration an

= \frac{1}{5} x^{5} arcsin (x) - \int \frac{x ^{5}}{5 1 - x ^{2}} d x

\int \frac{x ^{5}}{5 1 - x ^{2}} d x = \frac{1}{5} (- 1 - x^{2} + \frac{2}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}})

= \frac{1}{5} x^{5} arcsin (x) - \frac{1}{5} (- 1 - x^{2} + \frac{2}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} x^{5} arcsin (x) - \frac{1}{5} (- 1 - x^{2} + \frac{2}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}}) + C