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人気のある微分積分 >

integral from 1 to 2 of (x^2+7)/(3x-x^2)

解

\int_{1}^{2} \frac{x ^{2} + 7}{3 x - x ^{2}} d x

解

- 1 + \frac{23}{3} ln (2)

+1

十進法表記

4.31412 \dots

解答ステップ

\int_{1}^{2} \frac{x ^{2} + 7}{3 x - x ^{2}} d x

因数

\frac{x ^{2} + 7}{3 x - x ^{2}}

= \int_{1}^{2} \frac{x ^{2} + 7}{- x ( x - 3 )} d x

定数を除く:

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

= - \int_{1}^{2} \frac{x ^{2} + 7}{x ( x - 3 )} d x

拡張

\frac{x ^{2} + 7}{x ( x - 3 )} : \frac{x ^{2}}{x ( x - 3 )} + \frac{7}{x ( x - 3 )}

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}^{2} \frac{x ^{2}}{x ( x - 3 )} d x + \int_{1}^{2} \frac{7}{x ( x - 3 )} d x)

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

\int_{1}^{2} \frac{7}{x ( x - 3 )} d x = - \frac{14}{3} ln (2)

= - (1 - 3 ln (2) - \frac{14}{3} ln (2))

簡素化

- (1 - 3 ln (2) - \frac{14}{3} ln (2)) : - 1 + \frac{23}{3} ln (2)

= - 1 + \frac{23}{3} ln (2)

グラフ

人気の例

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