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人気のある微分積分 >

integral from 1 to 4 of (x^2+7)/(5x-x^2)

解

\int_{1}^{4} \frac{x ^{2} + 7}{5 x - x ^{2}} d x

解

- 3 + \frac{78}{5} ln (2)

+1

十進法表記

7.81309 \dots

解答ステップ

\int_{1}^{4} \frac{x ^{2} + 7}{5 x - x ^{2}} d x

因数

\frac{x ^{2} + 7}{5 x - x ^{2}}

= \int_{1}^{4} \frac{x ^{2} + 7}{- x ( x - 5 )} d x

定数を除く:

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

= - \int_{1}^{4} \frac{x ^{2} + 7}{x ( x - 5 )} d x

拡張

\frac{x ^{2} + 7}{x ( x - 5 )} : \frac{x ^{2}}{x ( x - 5 )} + \frac{7}{x ( x - 5 )}

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}^{4} \frac{x ^{2}}{x ( x - 5 )} d x + \int_{1}^{4} \frac{7}{x ( x - 5 )} d x)

\int_{1}^{4} \frac{x ^{2}}{x ( x - 5 )} d x = 3 - 10 ln (2)

\int_{1}^{4} \frac{7}{x ( x - 5 )} d x = - \frac{28}{5} ln (2)

= - (3 - 10 ln (2) - \frac{28}{5} ln (2))

簡素化

- (3 - 10 ln (2) - \frac{28}{5} ln (2)) : - 3 + \frac{78}{5} ln (2)

= - 3 + \frac{78}{5} ln (2)

グラフ

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