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

解答

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

\frac{11 π}{192}

十进制

0.17998 \dots

求解步骤

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

使用分布积分法

= [\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}

带入上下限计算后得:

\frac{11 π}{192}

= \frac{11 π}{192}

\int_{2 x}^{x^{3}} e^{- t^{2}} d t

\int_{3}^{10} \frac{1}{x ^{2}} d x

\int_{0}^{1} sin^{2} (x) cos^{5} (x) d x

\int_{0}^{1} y^{\frac{3}{2}} d y

\int_{0}^{55} x^{\frac{3}{4}} d x