integral from 0 to 1 of x^5arcsin(x)

Lösung

\int_{0}^{1} x^{5} arcsin (x) d x

\frac{11 π}{192}

Dezimale

0.17998 \dots

Schritte zur Lösung

\int_{0}^{1} x^{5} arcsin (x) d x

Wende die partielle Integration an

= [\frac{1}{6} x^{6} arcsin (x) - \int \frac{x ^{6}}{6 1 - x ^{2}} d x]_{0}^{1}

\int \frac{x ^{6}}{6 1 - x ^{2}} d x = \frac{1}{6} (- \frac{1}{6} x^{5} 1 - x^{2} + \frac{5}{6} (- \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))))

= [\frac{1}{6} x^{6} arcsin (x) - \frac{1}{6} (- \frac{1}{6} x^{5} 1 - x^{2} + \frac{5}{6} (- \frac{1}{4} x^{3} 1 - x^{2} + \frac{3}{8} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))))]_{0}^{1}

Berechne die Grenzen:

\frac{11 π}{192}

= \frac{11 π}{192}

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