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人気のある微分積分 >

integral from 1 to 2 of xsqrt(2-x)

解

\int_{1}^{2} x 2 - x d x

解

\frac{14}{15}

+1

十進法表記

0.93333 \dots

解答ステップ

\int_{1}^{2} x 2 - x d x

u置換積分法を適用する

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

拡張

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

= - (- \int_{0}^{1} - u^{\frac{3}{2}} + 2 u 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_{0}^{1} u^{\frac{3}{2}} d u + \int_{0}^{1} 2 u d u))

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

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

= - (- (- \frac{2}{5} + \frac{4}{3}))

簡素化

- (- (- \frac{2}{5} + \frac{4}{3})) : \frac{14}{15}

= \frac{14}{15}

グラフ

人気の例

integral from 1 to 3 of 1/(x^2+4)

\int_{1}^{3} \frac{1}{x ^{2} + 4} d x

integral from 1/3 to 3 of (2-2/(x^2))

\int_{\frac{1}{3}}^{3} (2 - \frac{2}{x ^{2}}) d x

integral from 0 to 4 of 1000x

\int_{0}^{4} 1000 x d x

integral from 1 to 2 of 1/(t^2+1)

\int_{1}^{2} \frac{1}{t ^{2} + 1} d t

integral from 0 to pi of cos(x)sin^3(x)

\int_{0}^{π} cos (x) sin^{3} (x) d x