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人気のある微分積分 >

integral from 2 to 8 of x/(x+sqrt(2x))

解

\int_{2}^{8} \frac{x}{x + 2 x} d x

解

4 ln (\frac{3}{2}) + 2

+1

十進法表記

3.62186 \dots

解答ステップ

\int_{2}^{8} \frac{x}{x + 2 x} d x

u置換積分法を適用する

= \int_{4}^{16} \frac{u}{2 ( u + 2 u )} d u

定数を除く:

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

= \frac{1}{2} \cdot \int_{4}^{16} \frac{u}{u + 2 u} d u

簡素化

\frac{u}{u + 2 u} : \frac{u}{u + 2}

= \frac{1}{2} \cdot \int_{4}^{16} \frac{u}{u + 2} d u

u置換積分法を適用する

= \frac{1}{2} \cdot \int_{4}^{6} \frac{2 ( v ^{2} - 4 v + 4 )}{v} d v

定数を除く:

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

= \frac{1}{2} \cdot 2 \cdot \int_{4}^{6} \frac{v ^{2} - 4 v + 4}{v} d v

拡張

\frac{v ^{2} - 4 v + 4}{v} : v - 4 + \frac{4}{v}

= \frac{1}{2} \cdot 2 \cdot \int_{4}^{6} v - 4 + \frac{4}{v} d v

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

= \frac{1}{2} \cdot 2 (\int_{4}^{6} v d v - \int_{4}^{6} 4 d v + \int_{4}^{6} \frac{4}{v} d v)

\int_{4}^{6} v d v = 10

\int_{4}^{6} 4 d v = 8

\int_{4}^{6} \frac{4}{v} d v = 4 ln (\frac{3}{2})

= \frac{1}{2} \cdot 2 (10 - 8 + 4 ln (\frac{3}{2}))

簡素化

\frac{1}{2} \cdot 2 (10 - 8 + 4 ln (\frac{3}{2})) : 4 ln (\frac{3}{2}) + 2

= 4 ln (\frac{3}{2}) + 2

グラフ

人気の例

integral from 0 to 2 of (x/2+1)

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

integral from 0 to pi/2 of 7sin(t)-2cos(t)

\int_{0}^{\frac{π}{2}} 7 sin (t) - 2 cos (t) d t

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

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

integral from 0 to 1.2 of x

\int_{0}^{1.2} x d x

integral from-2 to 2 of 16-8x^2+x^4

\int_{- 2}^{2} 16 - 8 x^{2} + x^{4} d x