integral from 0 to 1 of (sqrt(x)-x)^2

解

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

\frac{1}{30}

十進法表記

0.03333 \dots

解答ステップ

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

拡張

(x - x)^{2} : x - 2 x^{\frac{3}{2}} + x^{2}

= \int_{0}^{1} x - 2 x^{\frac{3}{2}} + x^{2} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} x d x - \int_{0}^{1} 2 x^{\frac{3}{2}} d x + \int_{0}^{1} x^{2} d x

\int_{0}^{1} x d x = \frac{1}{2}

\int_{0}^{1} 2 x^{\frac{3}{2}} d x = \frac{4}{5}

\int_{0}^{1} x^{2} d x = \frac{1}{3}

= \frac{1}{2} - \frac{4}{5} + \frac{1}{3}

簡素化

\frac{1}{2} - \frac{4}{5} + \frac{1}{3} : \frac{1}{30}

= \frac{1}{30}