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

解

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

\frac{11 π}{192}

十進法表記

0.17998 \dots

解答ステップ

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

部分積分を適用する

= [\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}

限度を計算する:

\frac{11 π}{192}

= \frac{11 π}{192}