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人気のある微分積分 >

integral from 1 to 5 of (x^2+7)/(6x-x^2)

解

\int_{1}^{5} \frac{x ^{2} + 7}{6 x - x ^{2}} d x

解

- 4 + \frac{25}{3} ln (5)

+1

十進法表記

9.41198 \dots

解答ステップ

\int_{1}^{5} \frac{x ^{2} + 7}{6 x - x ^{2}} d x

因数

\frac{x ^{2} + 7}{6 x - x ^{2}}

= \int_{1}^{5} \frac{x ^{2} + 7}{- x ( x - 6 )} d x

定数を除く:

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

= - \int_{1}^{5} \frac{x ^{2} + 7}{x ( x - 6 )} d x

拡張

\frac{x ^{2} + 7}{x ( x - 6 )} : \frac{x ^{2}}{x ( x - 6 )} + \frac{7}{x ( x - 6 )}

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}^{5} \frac{x ^{2}}{x ( x - 6 )} d x + \int_{1}^{5} \frac{7}{x ( x - 6 )} d x)

\int_{1}^{5} \frac{x ^{2}}{x ( x - 6 )} d x = 4 - 6 ln (5)

\int_{1}^{5} \frac{7}{x ( x - 6 )} d x = - \frac{7}{3} ln (5)

= - (4 - 6 ln (5) - \frac{7}{3} ln (5))

簡素化

- (4 - 6 ln (5) - \frac{7}{3} ln (5)) : - 4 + \frac{25}{3} ln (5)

= - 4 + \frac{25}{3} ln (5)

グラフ

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