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人気のある微分積分 >

integral from 0 to 2 of xsqrt(2x-x^2)

解

\int_{0}^{2} x 2 x - x^{2} d x

解

\frac{π}{2}

+1

十進法表記

1.57079 \dots

解答ステップ

\int_{0}^{2} x 2 x - x^{2} d x

平方完成する

2 x - x^{2} : - (x - 1)^{2} + 1

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

u置換積分法を適用する

= \int_{- 1}^{1} (u + 1) - u^{2} + 1 d u

拡張

(u + 1) - u^{2} + 1 : u - u^{2} + 1 + - u^{2} + 1

= \int_{- 1}^{1} u - u^{2} + 1 + - u^{2} + 1 d u

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_{- 1}^{1} u - u^{2} + 1 d u + \int_{- 1}^{1} - u^{2} + 1 d u

\int_{- 1}^{1} u - u^{2} + 1 d u = 0

\int_{- 1}^{1} - u^{2} + 1 d u = \frac{π}{2}

= 0 + \frac{π}{2}

簡素化

= \frac{π}{2}

グラフ

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