integral from 0 to 1 of x*arcsin(x^2)

解

\int_{0}^{1} x \cdot arcsin (x^{2}) d x

\frac{π}{4} - \frac{1}{2}

十進法表記

0.28539 \dots

解答ステップ

\int_{0}^{1} x arcsin (x^{2}) d x

部分積分を適用する

= [\frac{1}{2} x^{2} arcsin (x^{2}) - \int \frac{x ^{3}}{1 - x ^{4}} d x]_{0}^{1}

\int \frac{x ^{3}}{1 - x ^{4}} d x = - \frac{1}{2} 1 - x^{4}

= [\frac{1}{2} x^{2} arcsin (x^{2}) - (- \frac{1}{2} 1 - x^{4})]_{0}^{1}

簡素化

[\frac{1}{2} x^{2} arcsin (x^{2}) - (- \frac{1}{2} 1 - x^{4})]_{0}^{1} : [\frac{1}{2} (x^{2} arcsin (x^{2}) + 1 - x^{4})]_{0}^{1}

= [\frac{1}{2} (x^{2} arcsin (x^{2}) + 1 - x^{4})]_{0}^{1}

限度を計算する:

\frac{π}{4} - \frac{1}{2}

= \frac{π}{4} - \frac{1}{2}